There were quite a few concerning things that Trisha Paytas said about David Dobrik and the vlog squad, and what of my personal concerns after watching a lot of different YouTubers and as they continue to grow and keep trying to top themselves. One of my concerns is what happened to Logan Paul when he went to Japan in that forest. So I'm really worried about where David Dobrik and the vlog squad is headed. What is up everybody? This is Chris from the Rewired Soul where we talk about the problem but focus on the solution. And if you're new to my channel, my channel is all about mental health and what I like to do. I like to pull different topics from the YouTube community and try to teach you how to improve your mental and emotional well-being. So if you're into that stuff, make sure you subscribe and ring that notification bell. So before I get started talking about why I'm worried about David Dobrik and the vlog squad and the whole YouTube machine that people get caught up in. Yes, I know the vlogs were deleted. Trisha Paytas deleted her video. I get it. I understand. But anyways, I do want to send a reminder and shameless plug my audio book for Rewire Your Anger is out now. And some of the stuff I talk about is how to manage your anger so you don't do something that you're going to regret. So if you struggle with anger, if you do things that you regret and wish you could take back, go ahead and check out the book. It's $7.99. It comes bundled with the e-book.  And the next thing I want to address real quick, a lot of people are saying to talk about what Trisha Paytas said right here about Brandon Calvillo. There's people dating Brandon in their friend group was dating a high schooler last year. She was 17 years old. And when I would talk about this with like David, he would literally be like, oh, it's a big deal. He's 24, like she's 17. Like, you know, just because it's the law like a year, like a six months and six months she'll be 18.  Like people have found this out and like Brandon will call David and be like, oh, like you got to delete these comments about her age and stuff like that. Because she was in high school when they started like dating and hooking up like it's this whole like weird thing.  And that's what I was trying to explain to Jason like that's an evil person. All right. So I'm not really going to address it like he may or may not have done this. Like I'm not here to like speculate about that stuff. There's no evidence. No young woman has come forward and said, yeah, I was underage dating Brandon. But if if this happened, then yeah, that's clearly messed up and it should be addressed. All right. Because obviously there is a power dynamic. And I guess one of the first topics I'll talk about like I've been watching David Dobrik for not as long as most of you. But Tristan got me into David Dobrik and I've been watching him. And I see that, you know, some of the guys like Dom in particular in recent vlogs, he's been sleeping with a lot of women, right. And, you know, some of them are sleeping with like fans and things like that. And they joking and have these stories.  And it's not very equal. So, you know, like if if women want to sleep with these dudes, go for it. Do yo thing. But that's one of the reasons that I am concerned, you know, one of the reasons I'm concerned about David Dobrik and the vlog squad is the influence that they have on people. Like, I'm a grown man. Like I'm an adult. I'm 33 years old. I watch them. And to be honest, like it reminds me. It reminds me a lot of my younger days when I was when I was a full blown alcoholic. All right. Like I see what they're like, I see what they do and they get messed up and they like mess around and goof off. And it looks like people are getting hurt and stuff and they laugh about it and all that like that's a lot of the stuff I did when I was in my early 20s. So, can I fault them for that? I don't know. Like is it dangerous? Of course it is. So, like the influences that they're having on young people, like basically one of the things that and like when I make these videos there for you and I want you to monitor what you're watching and how it's influencing you. Like watching some gigantic YouTubers doing this, it's going to justify your behavior, right.  Right now is that if David Dobrik or any member of the vlog squad goes to jail, they will be out very quickly.  And unless your parents bail you out, you're probably going to stay there for a while. All right. So I've made videos recently about things like, you know, social cognitive theory where, you know, we've seen how what people watch can influence their real life. So that is one of my concerns. And as I mentioned, me being an alcoholic in recovery, I've been sober for about six and a half years. Hi, my name is Chris. If you haven't met me yet, I'm a recovering alcoholic and drug addict is that, you know, there's not enough out there to really know if they're struggling. If any of the members are struggling with alcoholism. But like Tristan, wasn't it Zane who like went to the hospital or something like that? Yeah. Yeah. Yeah. Yeah. Oh yeah. Yeah. She reminded me that they walk through fire and things like so something that you guys need to understand like part of what I do, you know, working in addiction treatment and just trying to be an advocate for mental health and addiction is, is that you need to understand this. Okay. So on average, and I haven't seen a new report yet, but on average about over 70,000 people died from the opioid epidemic in the United States. All right. So people look at that number. That is the number that is just shoved in your face. But there are tens of thousands of people who die from other substances. Now, one of the numbers that is always, always skewed is how many people die from alcoholism. And here's why most people do not die from alcohol poisoning. It's possible, but it's not happening all that much.  But what I mean by the statistics are skewed is that most people who die from alcoholism, it's alcohol related accidents. All right. It doesn't take much for you to, you know, do a quick Google search and hear about like college students who you know, we're going through like, like, they were just like binge drinking or something like that. Someone drowned in a lake or whatever. So those stats don't necessarily go to deaths because of alcohol. And I hope that makes sense. You got to think about drunk driving accidents and all of that, right? But the other number that isn't really included in those statistics is health problems as a result of alcoholism. Alcohol is one of the most dangerous substances when it comes to what it does to your body. Like you can actually die from alcohol withdrawal, but it can cause issues with your heart, with your liver, with your kidneys, all sorts of stuff. Okay. Like when I first got sober, I had congestive heart failure. That's one of the reasons I followed up with my doctor. So like that was six and a half years ago. And thank God my heart is in much better shape now, but alcohol was killing me based on how much I was drinking. So one of the things is that alcoholism isn't really looked at as this like disease of addiction type thing. When people think of addiction, they think of like, you know, heroin or prescription medications. And, you know, since alcohol is like the most legal substance out there, it's very easy to justify it. So you look at David Dobrik, you know, the vlog squad and what they're doing. Like it's easy to say, oh, we're just yelling and partying and stuff like that. But we don't know if anybody has a problem and I don't want to speculate that. But again, I want to talk to you about that. So if you're somebody who is obsessed with the David Dobrik vlogs because you can relate and you and your friends like to be all wild and zany, like it's important to look for the symptoms of possible alcoholism. Because when we see all of the stunts that they're doing and the things that they're doing for the vlog, like it's possible that someone could die because like that's literally what Trisha Paytta said in her video, that she's worried that somebody is going to die. All right. So the last thing I want to talk about is this whole back and forth that we're seeing right now, the videos got deleted and everything like that. But you see, you know, when like I really felt for Trisha when she was, you know, crying in her video, because they pressure Trisha into doing things, but they pressure each other into doing things and say, it's for the vlog, it's for the vlog, it's for the vlog. Like, trust me, what Logan Paul did was sick, was disgusting, but like I understand where his headspace was. Like when you get caught up in this YouTube machine, as these big YouTubers do, you're trying to, you're trying to one up yourself. You're trying to do bigger, bigger, bigger, bigger things. I'm like, where does it stop? And more people, more people often than not, and this is all of you as well, you keep pushing the limits until it's too late, right? Logan Paul learned that lesson and when I see David Dobrik in the vlog squad, I'm like, dang, I don't think they're going to learn that lesson until it's too late. Now, do I think they're going to go to Japan and do something stupid? No, but I think we're going to see something on par with it. You know what I'm saying? Like, this is going to be something that I like just, I've seen people, you know, go down this path and something that I'm doing as a YouTuber and watching the YouTube community is you see trends, you see patterns, you see all these things. And like, they are definitely on that road. So I don't know, maybe Jason being the grandpa of the group. That was, that was, that's funny though. That's funny. Tristan's laughing. You can't see her. Like, maybe you just say, yo, let's dial it back a little bit. You know what I mean? But there are bigger things to worry about rather than, you know, the views and all of that. And something that I always say, like, you know, we got to check our motives. We've got to check our intentions. Like, why are we making videos? Why are we doing this? Like, there is definitely, you know, a need for, you know, David Dobrik in the vlog squad because it's entertaining, right? But it's like, at what cost? You see what I'm saying? So those, those are kind of my thoughts on that.  Like when I'm talking about how you keep pushing the limits until it's too late. Like, are you overworking yourself?  Like, this is a thing. Everybody needs therapy. Everybody needs therapy. The problem with most people here, like, let me let, come here, come here. Let me let you in on this secret. The problem with most people is that they don't go to therapy until they freaking, they snap. You know what I mean? Like, you need to say, whoa, whoa, whoa. Like the, the wheels are getting a little bit wobbly. I need to get in there now. So check in with yourself. Don't keep pushing yourself until it gets to that breaking point. Early interventions are key when it comes to mental health. All right. Talk to your doctor, talk to your insurance company, find a therapist. There's always a link down below for better help online therapy, which is a service that I personally use. Tristan's used it. A lot of people use it. All right. But anyways, that's all I got for this video. If you liked this video, please give it a thumbs up. If you're new, make sure you subscribe and ring that notification bell because I make a ton of videos. And a huge, huge thank you to everybody supporting the channel over on Patreon. You're all amazing. And if you would like to help support the channel and get exclusive perks, discounts, free books, all that stuff, click or tap right there. All right. Thanks so much for watching. I'll see you next time.


Say guys the contacts on these points need cleaning Let's clean them.  Now yours may not have that switch It may have one up at the pump for you may have to shut your circuit breaker off If you don't know which one it is just shut your main breaker off It's not going to take very long. So don't worry about that. Just go and shut your main breaker off in the house and That should take care of it. So now let's go out here and make sure We're going to go to the points and make sure that The power is turned off Now I'm going to take this off and I'm going to push these points together Okay, it's not running. So the power is off Okay, so now let's clean these points All I've got here is a little piece of sandpaper It's a 150 grit and I'm just going to take that Stick that down in there. I'm gonna flip the points closed And I'm just going to drag it through a couple times just like that. Can you see that? You don't want to go too much Just to get that black stuff off of there And you can see say it's cleaning up pretty good.  All right, we're gonna watch watch the points. We're gonna turn this to pick it on right here And we're gonna make sure when it loses pressure This points kick on there. They go now. We'll make sure they kick off and they did so we're set. We're good to go guys awesome now Just want to stick our little deal Back on top of here. We're good to go And guys, that's all it takes to clean your contact points your breaker points on Your water pump they should be now yours might be look a little different or whatnot But it's just all pretty much the same Finding points there's they're gonna be right there somewhere easy to get to find them and just clean them up and check that and Remember make sure you shut your power off before you go fiddling around with that so you don't Get zapped and everything should be good. So guys remember later this evening 9 o'clock Q&A. All right. Have your questions ready. All right guys. Thanks for watching. Shea bear the myth main a legend a mile for now Bye-bye guys. Take care


Hey everybody, it's Eric Johnson from Airtate Thrust Nation.  So what are we going to do? We have basically three options. We can be depressed, bummed out that the season's gone and do nothing. We can try to maintain or we can come out of this better and I think that's really what the goal should be. I know it's hard to say try to stay normal because you know what, it's not normal. The thing that we want to really focus on is getting better, coming using this time wisely. There's so many things that you can be doing and should be doing and we've got a nice series of stuff coming. We're going to show you a quick video on one of our athletes who trained a few weeks back from out of town. We're going to kind of show you inside a training session. There's some of your normalcy, but remember to use this time to study to get ahead to come out stronger and that's going to be the goal for all of us to keep our sanity. So you guys stay healthy, safe, check out the video. Not bad. Yeah, now you're already on top of the axis. Look at this line. So today I had a training session with a young guy named Alex Paulson in from the Chicago area and so again we went through and you'll kind of notice how we made the progression changes. He doesn't reverse because largely because the positions don't allow him to. He started to make some pretty significant changes, speed started to increase. He was throwing at his PR. Okay, so the first thing that we focus on when we take any new athletes, we're going to go through and what you didn't see is how we build the athlete from the start of the throw and we kind of just let him do what they do. We don't start coaching right away. I always want to have an athlete come in, show me what they're doing, show me how their routine works, get an idea of how they move and then from there we'll start to build through the throw. So at this stage in the video, what we're going to do is show you how we'd actually already gone through the beginning pieces of the throw, built it up and now we're starting to explain how the chain reaction works and that we're going to be applying that to the entire training session. So we think one of the big important things for coaches and throwers is obviously to be able to show, as a coach we're showing what they're doing wrong as well as what they're doing right so we can try to really relate the difference and you're going to see here as we kind of build through the athlete's throw. So one of the important things is when you're doing a, when we're moving, obviously setting up alignment, setting up the positions and then setting up the chain reaction is going to be really critical to create changes for the thrower. You know, one of the, one of the important things that we do is when we're looking at a beginning throw versus an advanced throw, again again you're going to notice that the gross movement is actually always pretty close and it's the details that are going to make a huge difference. Those are the things that we talked about, setting up alignment, setting up a chain reaction and then obviously we'll do some things and you can see we'll compare to more lead throwers so it really shows those differences and again you'll see that again it's close but those differences make massive differences in how far you're going to throw and when you do that day after day it's going to result in a big thing. This is why we're typically able to see pretty significant results with our throwers in a matter of, you know, 90 minute training session. So one of the things that we wanted to do is give you a little bit more behind the scenes. Now this athlete, it's the first time we've ever worked with him. He's got like a good squat but you can see when the athlete was standing the athlete would stand with his legs kind of hyperextended in a natural position. That's a good indicator that the hamstring and quad ratio is up. That's going to cause a whole bunch of problems in how the athlete moves. The athlete was having a hard time getting loaded and moving through the position so moving, he's carrying extra body weight that we kind of refer to as just not usable good functional weight and so getting himself in better condition will also help and that's a real important thing for a lot of you throwers. And we are on week two of this coronavirus mayhem and it's just totally changing everybody's life, routines, jobs, it's devastating. And so throwing in perspective of people's health or sports, it's not to override it but I think at the same time trying to keep your routines as normal as possible I think is a good thing. We're providing our online members with suggestions and workouts and things that they can do in their driveway, in their backyard, in their basement, kitchen, whatever. Try to stay safe, healthy, try to stay as much to a normal schedule as possible and thanks so much guys, we'll see you on the next video.


Hey there, chameleons! It's Camille, and today I'm going to show you how I install a ponytail into my curly hair without using any heat at all. So the first thing you need is obviously a ponytail. This is called the Maya from I&H. It's a synthetic 26-inch ponytail, and I did a video last week on my channel where I showed how I dyed it green since it is synthetic hair. You can't just use regular hair dye, so I showed how to dye synthetic hair to match my regular hair.  You could dye that or not. If you want to do a synthetic one, pick one that's your hair color. If you can't find one, then you can dye it like this, but that video will be linked in the description box down below. So I try not to use heat in my hair as much as possible because my hair is bleached and dyed so it's already kind of fragile, so this is a method that I figured out on my own because I was trying to hack the system and that I have to straighten my hair to wear a straight ponytail, so I figured that this could be kind of useful to any other curly girls or any wavy or whoever who doesn't want to straighten their hair to match the ponytail. The first thing I did was get the ponytail ready. I sprayed it down nice and wet, and then I used my synthetic hairbrush to just comb it out so it's nice and detangled. I feel like it gets tangled so easily, so I was really careful with that. And then the other most important thing is to prep your hair, and the way that you're going to make this work is to prep your hair properly. So what I did is last time I washed my hair, I washed it, I conditioned it, and then I pulled it back into a ponytail, so instead of letting my hair down and just dry curly from the root, it was kind of pressed back and had to dry in more of a straight way, so you can see that this hair is a lot straighter than if I had my hair down and curly. You can see in my pictures, my Instagram, my videos, whatever, and you can also see here, if I take down my hair, it's gonna look so crazy, but this part was let to dry normally, so you can see that it's all curly, but I let it dry all together, so instead of having individual curls that come out when having the volume, I just had it as much together as possible to kind of consolidate the curls, so that's the first part is letting this part of your hair dry straight and then kind of consolidating the rest of your curls, and depending on how long your hair is, you're gonna do this a little bit differently. If you really have super short hair, then you can get away with just doing a little bun and then hooking in the ponytail, wrapping it, and you're good. My hair is kind of like medium length at this point, so I'm gonna have to do it a little bit differently, and if you have long hair, then you can either get a ponytail that is the same color and texture, and then just leave your hair down in it. If you don't have the same texture, you could straighten it or blow dry it, and then just like do that, or you could try to do what I'm about to do, but it might add like a lot of volume up there, so it just depends on how long your hair is, but this is perfect for medium and then short hair is even easier.  This one is super easy to use, and you can find it online. I'll link everything in the description box down below, and then some Eco Styler gel or any gel that has somewhat of a medium hold and is kind of malleable, and then you're gonna want a boar bristle brush. This one is from the brand Diane, and it's pretty soft on my hair, and it also helps really smooth it out, which is great for making it like a pure kind of straight, so definitely recommend a boar bristle brush. Oh, and you need a hair scrunchie. So normally I use satin scrunchies in my hair. They're super loose, they're super slippery, they don't add any friction in my hair. It's nice to not have any breakage or any tugging, but unfortunately for this ponytail, you can't just put on loose hair, so I'm gonna use a strong hair scrunchie for this one. First I'm gonna take my hair down, and I'm gonna use the spray bottle and wet it, and you don't want it to be soaking because your hair is less strong when it's super wet, so you just want it to be wet enough that you can brush through it and add the gel. I'm also gonna spray the curly part a little bit. I'm also gonna get the middle because I want that part to stay flat when I pull my hair back and not have like weird extra volume. Then I'm gonna use a nice little glob of the Eco Styler gel. I'm gonna spray my hand so it's a bit wet, then I'm gonna mix it all up in my palms, get that nice ASMR. I should have probably taken all my rings off before this, but we'll get into that later. Now I'm just spreading the gel along all of the outside parts that are going to be pulled back when I put in the ponytail. You want to make sure there aren't any lumps underneath so that it can have a nice smooth spread. I don't do this style that often because I don't like having so much product in my hair, and because the ponytail like pulls on your head. So this is just a really cute look that I'll do sometimes, and I really like it, so I think it's worth it. It's pretty slick right now, so next I'm going to use the boar bristle brush, and I'm going to make sure that I smooth out everything, and just get it as smooth as possible. Once you're happy with how slicked down your hair is, the next part is super important, and you want to make sure you get this right before moving on because you cannot go back and change this. You want to find the perfect position for your ponytail, so I'm doing a high pony, and I like that to be about this high on top of my head. If you're doing a lower pony, then obviously you go lower, or you just position it wherever you want it on your head, and then you also want to make sure that it's centered because you just know and do this whole thing, put in the ponytail, and then realize that it's like lopsided. So that is a super important thing to check before you put the scrunchie in, so I'm lining it up with my widow's peak because that's kind of like a perfect indicator of where the center of my head is, and then once you have it you want to do one last test, and this is something that I realize is really important because I've definitely made this mistake before. If you're going to put in a really tight scrunchie or ponytail or whatever, you want to move your head around first to see if it pulls on any of your hair, because I've done it super tight before where if you move your head down these hairs it just pulls so hard that it hurts and you can't move your head down. So you want to tie it up in a way that is centered and not painful for when you're moving your head around, so once you have found that position use a nice and strong hair scrunchie like this one, and tie that up. Normally I use satin scrunchies for my hair whenever I'm doing something, but unfortunately you can't do that for a tight look like this, so I'm just using a regular hair scrunchie. So I'm happy with this position right here. It's centered, it's pretty tight, and my hair is nicely slicked back, so now it's time to divide my hair into three sections, and this just depends on how long your hair is and how thick your hair is, but I am doing three equal sections like this, and I'm going to do a braid for each of these sections. If it makes it easier you can use your water spray to wet down your hair to make it easier to braid, and then you can use gel or any other thing that you would braid your hair with normally, and it doesn't have to be like perfect sleek braids. If you do want that then you can wet your hair all the way and do braids like that, just whatever kind of braid that you like. I'm just gonna do some really simple easy braids that don't really hide my texture that much. I'm not really super concerned about this, but if I was doing a super sleek look maybe I would do it while my hair is wet and do tighter braids. I'm also adding a little bit of avocado oil to make my hair a little bit easier to braid and a little bit smoother, or if I was doing more of a rough tassel look I would do it a less precisely, but this is what mine looks like in the end. But this is perfect for medium length hair so that's why I'm doing it like this, and I feel like a lot of you guys have medium length and curly or wavy hair, so this is kind of the in-between catch-all that's pretty easy for like most hair types.  So different ponytails are made differently but the one that I have has a little clip in the back and then it has a velcro strip with this little hair section that you use to wrap it around the base of your ponytail. So we're gonna go in and start with this clip in the middle like right above the hair tie. So the hair tie is right here and we're gonna put the clip starting here and going in right so here's the clip and I'm putting that right here pushing that in making sure it's in nice and firm and there we go. You can feel the weight and it's not pulling it's not super heavy so that is a good sign. So now I have these four things sticking out three of them are my braids and one is the little velcro thing that you wrap around so I'm pulling out two braids on this side and one on this side so they won't get wrapped up in the velcro. So I'm pulling down the velcro section here and then flipping my head and looping this around so that's on nice and tight and normally if I had really short hair and I just made a little bun I would hook that in and then I would just wrap this part around like that and that would be it but because my hair is a little bit longer than just a little bun and because my hair is a tiny bit different than the color of the ponytail I tried to get it as close as possible but it's not exact but I figured that this would be a really cute way to kind of fade the colors into each other and also add a little detail with the wrapping section. So I'm wrapping this straight one all the way around at the bottom and then I'm gonna pin that in place then I'm separating out my three braids and I'm going to start wrapping those around also on the outside of the ponytail then I'm pinning each one in place now that this is all done the last step is to do edges so I'm going to be using my got-to-be extra glued gel and this is a lot stronger than the Eco Styler gel so this is what I like to use at the end to lay my edges and then I also put a little bit on the rest of my hair to just make sure that it's as sleek as possible and I'm using my little edges brush that I got on Amazon it's so cute it has the brush and then it has the tiny little comb so this is perfect for doing both the edges and then like a little bit of smoothing on the top of my head I'm just putting a little bit on my fingertips and I'm just gonna put chunks of it all the way around my hairline and then I'm going to use my brush and smooth that out then I'm using a bit more and I'm just going to smooth that out a little bit on top of the rest of my hair so I got some edges to bun now I'm using my little brush and I'm laying down very very basic edges and smoothing it all the way back I usually kind of avoid my widow's peak and then start adding those swoops far in the rod again. The last thing I'm gonna do is I'm going to use my brush and just brush very gently through this because when I move around it gets a little bit tangled and I feel like this gets tangled super easily so I just want to have it nice and sleek at the end. Another thing you can do is if you have that section on the ponytail that sticks out for the wrap around you can braid your actual hair into that so it will help kind of with more of a smoother gradient but I didn't do that it's just another thing that you could do and voila you have a perfectly installed ponytail and it is serving today I love this ponytail I've worn it twice so far but I just cannot get enough of it and I really like the pictures that I took in it and just overall I'm so happy with this and very happy with this look so the next time I dye my hair a new color I'm definitely going to be getting another ponytail and dyeing it myself and seeing how close I can get it to match the next hair dye color so I'm very excited about that and I hope you guys found this video helpful I used absolutely no heat in my hair and I was able to get it installed and looking pretty seamless so if any of you guys are looking to do that with a straight ponytail and you don't want to damage your hair any further hopefully this video will help you out with that all the products that I use are linked in the description box down below as well as all of my social media so don't forget to follow me on there I hope you guys all enjoyed this video thank you so much for watching I wish you all a happy healthy life and I will see you in the next one


The Appalachian Advantage plan is the simpler, better way to get the phone you actually want instead of paying hundreds of dollars upfront.  It's called the Appalachian Advantage and is available at Appalachian Wireless. As the summer season dawns upon us, many pools will be reopening, including some around our region. Monday, Mountain Top spoke with Prestonsburg Park Director Richie Schoolcraft about Archer Park's pool and when we can expect to see it open. The pool opens this season Saturday at 12. We're excited to open up and we'll be open through the weekend for Memorial Day. We'll be closed in the middle of next week though. All of our lifeguards go back to school and they'll get out of school next week and there's a graduation next week and we'll reopen again the following weekend for the rest of the summer. Over several weeks of pool preparation, Schoolcraft mentions that the pool is seeking various forms of employment such as lifeguards for the summer. Along with the pool's opening, Schoolcraft mentions the opening of Dewey Lagoon, which is expected to be open this weekend. Saturday we're going to open up at the old Swingpool area. We have a 100-foot beach area and 100 feet of inflatables and that would be it's new this year and so we're doing all the finishing touches this week to open that at the same time. For employment or if you have other questions you can call<PHONE_NUMBER>. For Mountaintop News, I'm Joel Kordjall.


Welcome back to our lecture series Math 42-30, abstract algebra two for students at Southern Utah University. As usual, be your professor today, Dr. Andrew Misseldine. So we're reaching basically the end of our lecture series. So previously in lectures, well, earlier in the 30s, we developed this idea of field theory, Gaoua theory culminated with the fundamental theorem of Gaoua. We competed some Gaoua groups, and then finally we proved the insolvability of the Quintic polynomial equation. Some pretty big notions here. So in these last three lectures for Math 42-30, what I want to do is basically do it all over again. But not in the complete depths we did before. So what do I talk about doing it all over again? If you look at both of these lecture series Math 42-20, abstract algebra one and 42-30, we studied in great depths the ideas of groups, rings, and fields. What I want to do is do it all over again, but basically now in an alternate universe, where instead of the group being the most fundamental algebraic notion for which we build on top of that, what is a ring? Well, a ring is just an abelian group with a second operation that behaves well with the first operation, which we'll call them addition and multiplication. And then a field is just a very highly structured ring. I want to do this over again, groups, rings, and fields, but in an alternate dimension, a bizarro world, the multiverse, so to speak, for which this time instead of developing the theory of groups, we're going to talk very briefly about the idea of a semi-lattice. In many ways, semi-lattices behave like groups, but in many other ways, they're quite alien objects. And then building upon that same theory, if we take a semi-lattice and add a second operation, we get the equivalent of a ring, what we call that a lattice. And then if we take a highly structured ring, what's like the perfect ring, well, that's going to give us a field as we did before, but in this context, we're going to lead to the idea of abelian algebra. And so very briefly, in the last few lectures of this lecture series, that's what we want to do, have a parallel between groups, rings, and fields in this setting of lattices. So the analog of a group in this alternate universe is what we call a semi-lattice. So a semi-lattice is a set equipped with a binary operation. So we take two elements, we combine them together and produce another element of that same set. It's going to have three axioms and some of the axioms are going to feel similar. So for a semi-lattice, our first axiom is going to be associativity. That's exactly what we took as the first axiom of groups. So just as a reminder, if we take three elements of our set, say the set is L, you take the elements X, Y, and Z, if you then take your binary operation, this, we could call this multiplication if we still wanted to, but I do want to make a look a little bit different than what we had from group theory. Also, I'm trying to generalize a different algebraic structure. What is the algebraic structure I'm trying to go through, right? With groups and rings and fields, we try to capture the notions of addition and multiplication. But as we study lattices in these lectures, we're not trying to capture the algebra of addition and multiplication. We're actually trying to go for the algebra of unions and intersections. I mean, at the very beginning of math 42-20, we spent time talking about the algebra sets. That is, we studied the algebraic structure of sets. So we have unions and intersections and set complements and things like that. And what we want to do now is generalize that concept to the algebra of order. Can we come up with an algebra unordered things, partially ordered sets, generally have algebraic structure and that's what we want to axiomize in this setting right here. Thus, with this semi lattice, our symbol here kind of looks like a union symbol, although it comes to a point. This symbol right here is commonly referred to as a join. A join sometimes it's referred to as a V. Like literally VV, you know, like V mon did evolves into V in that situation. I'll typically call it a join. That's a very common term. Sometimes people call it the least upper bound operator and we'll make some more sense of that by the end of this video. So we have our join operation and it's associative. So if you take Y join Z and then you join X onto that, that's the same thing as X join Y join Z. All right, so we do require this operation be associative. Our second axiom, slightly different approach to what we did with groups is our second axiom actually will be commutivity. So for any elements X and Y inside of the set L, we require that X join Y is equal to Y join X. We actually are gonna build commutivity into the cake here, cook it into the cake. For groups, that was sort of like, yeah, you could be commutative, you get an abelian group. But for simulatuses, it is required to be commutative. But again, that's not too alien from groups because we did study a lot about abelian groups. Commutivity was a good thing. It's our third axiom that really is gonna be deviant from what we expected with groups. The last axiom is idempotency. That is we require that every element be idempotent. So what that says in this context is if we have some element X inside of our set L, we have that X join X is equal to X. That is if you take an element and operate by itself, you always just get back the original element. Now this is a very alien concept from groups because in a group, the only idempotent element is the identity. And so in this respect though, we require for simulatuses, we require that every element is idempotent. And the poster child of a simulatus is exactly that of a power set with unions. So that is if you take any set whatsoever, consider it's power set, the set of all subsets, and you would join the power set with the union operator, this then forms a simulatus. Union is associative, it's commutative. And if you take any set, A union A, you just get back A. This gives you in fact a simulatus. We could also have done this with intersections. If you take the power set of X and adjoin to it the intersection operation, this is likewise a simulatus. And this is actually where the name simulatus comes from. It actually seems to suggest, I mean simi here kind of means like half half of a lattice with a lattice, a lattice which we'll define later on in this lecture, not in this video though. A lattice will be the structure that we get when we put these two things together. When we take unions and intersections together, it forms a lattice. So right now, if we only consider one of the operations, we get a simulatus, half of a lattice, all right? And so a simulatus is trying to generalize the structure of the union operation, much like how groups want to generalize addition and multiplication as algebraic operations. Simulatuses wanna generalize the notion of unions and intersections. But these are not the only type of simulatuses out there. For example, take the set bracket in to be, we're gonna take this to be the divisors, the, excuse me, the positive divisors. I want that to be the case here. We wanna have the positive divisors of some positive integer in like so. And then if you take this set in bracket and you equip it with the divisibility, the divisibility, excuse me, not divisibility. I'm getting a little ahead of myself there. Divisibility is not a binary operation. It doesn't produce an element, it's a relation. What I wanna do is actually, excuse me, the GCD, GCD in that situation. This also forms a simulatus for which, for example, you could take the divisors of 12. So you have 12, six, four. Let me scoot this up a little bit. We're gonna get three over here, two over here, one. Whoops, there's no line there. You get this picture right here. And so if you look at the divisors of 12, and you can take the operation of GCD. So like for example, if you take three and two, their GCD is one. If you take six and four, their GCD is two. If you take six and three, their GCD is three again. This gives you a binary operation, and GCDs will in fact be associative, it'll be commutative, it'll be idempotent. If you take the GCD of six and six, it's still six. Sorry, no, that's sort of like the devil's number there, 666, but it's just a mathematical example. We can create a simulatus using GCDs. Alternatively, you also could create a simulatus using LCMs, the least common multiple of these things. This would likewise be a simulatus. And this is to suggest why we call these things simulatuses. Like I said, simulatuses is half of a lattice. Why a lattice? Well, when you look at these Hase diagrams, which some people call Hase diagrams, lattices. The Hase diagram is basically the algebraic structure, or excuse me, a lattice is the algebraic structure of this type of partially ordered set. You can create a simulatus from a Hase diagram from a partially ordered set when you have a well-defined greatest common divisor or least common multiple. That is when you have a least of our bound or greatest lower bound on a partially ordered set. And that's actually what I wanna prove right here. This is the main result for this video on simulatuses is that the construction, the example I just mentioned is basically the only example of disomorphism. Which don't get me wrong, there's a lot of diversity when it comes to simulatuses. It's very easy to come up with them because we can just draw diagrams. We'll do that at the end of this video. But in the meanwhile, every simulatus has a natural partial order structure onto it. And therefore, the simulatus is capturing the algebraic structure of that partial order and vice versa. So imagine we have a simulatus, L with the symbol join, it's a simulatus. We can then define a partial order on L. That is we say that X is less than or equal to Y if and only if the product X join Y is equal to Y. This defines a partial order on the set L. So L is naturally a poset, partially ordered set. And in fact, the element X join Y is necessarily the least upper bound for X and Y with regard to this partial order. All right, so to prove it to partial order, there's three things we have to show. It has to be reflexive. It has to be anti-symmetric. It has to be transitive. So that's what we're gonna start with. So assume that L is a simulatus and consider the element X. Well, by the idempotency axiom we have that X join X is equal to X. Well, when we look at the definition of what it means to be less than, that's exactly it. X is gonna be less than or equal to itself. So this is in fact a reflexive axiom, a reflexive ordering, a reflexive relation. So in particular, idempotency directly implies that this thing is reflexive. I want you to pay attention to these things. Three axioms of simulatus, three axioms of the partial order. So let's prove that it's anti-symmetric. Take elements X and Y inside of the lattice, simulatus, and let's suppose that X is less than Y and Y is less than or equal to X. Well, if X is less than or equal to Y, that means that X times Y is equal to Y and if Y is less than or equal to X, that means Y times X is equal to X. But this is a commutative operation. So X by this assumption is equal to Y times X, but since it's commutative, this product is equal to this product X times Y, but X join Y is equal to Y by this assumption and so we get that X equals Y and so our relationship is anti-symmetric. So this time we see that the commutivity axiom implies the anti-symmetry axiom. So this is one-to-one correspondence between axioms coming into play here. So maybe it comes for no surprise here that the transitivity property of the relation will follow exactly from associativity. That's what we're gonna see here. So let's suppose we have three elements, X, Y, and Z that belong to the simulatus and suppose that X is less than or equal to Y and Y is less than or equal to Z. Well, since X is less than or equal to Y, that means that X times Y is equal to Y and since Y is less than or equal to Z, that means Y times Z is equal to Z. Now we're gonna use associativity here. So what we need to show is that X join Z is equal to Z. I would show that X is less than or equal to Z. Well, X join Z, because Z is equal to Y join Z, we can substitute that into the equation. So we get that X join Y join Z is then equal to X join Z. But then by associativity here, aha, associativity is gonna give us this equality right here. X times YZ is equal to XY times Z. But then by the previous assumption, XY is equal to Y, we make that substitution. And so we get that XY times Z is equal to Y times Z. But then also by previous assumptions, Y times Z is equal to Z. And so that then gives it to us. X times Z is equal to Z, so therefore X is less than or equal to Z. This is a direct result because of the associativity axiom. So we get transitivity. There's this direct relationship here. Idenpotency, implied reflexivity. Commutivity implied anti-symmetry and associativity implies transitivity. So we get this partially ordered set here. Every simulatus has a natural partially ordered set relation. Now I do wanna make a comment here that the way we defined it, we said that X is less than or equal to Y if and only if X join Y is equal to Y. This is not how everyone works with simulatus. Some people go the other way around. Some people will say alternatively that X is less than or equal to Y if and only if X union Y equals X. So they go the other way around. Compared to what we've developed here, this would be the reverse relation, the reverse partial order. Given any partial order on set, you can always turn them around. And so for our purposes, this is what we would say is greater than or equal to. So when it comes to simulatus and lattices, sometimes they have the orders flip around. This is particularly the case with simulatuses. So don't lose too much sleep over such a thing. That is the case. Now, personally for me, when you define it like this, you use the symbol join to describe it. In this setting, really what you should say is that X is greater than or equal to Y if and only if X, well, let me back up. If you wanna say it the way that I had it written before, X is less than or equal to Y. In that setting, you'd actually say that X join Y is equal to X. Sorry, not join. This symbol right here is what we call the meat. Or some people call it, what else do people call it? Sometimes they call it a smash. We'll stick with meat in this situation. Wedge, that's a term that some people use here, a wedge. That was the word I'm looking for. Some people actually you call this symbol wedge. So you have to be very careful. But in latex, this symbol right here is backslash V. V-mon, digi-volve two, like I said before. And this symbol is the wedge symbol in latex. So some people call them that as well. We will typically call these meat and join. This is join, this is meat. When it comes to a similatus, there's really no distinction between the two. The two can be equivalent to each other. But when it comes to a lattice, we will distinguish between what a meat is and what a join is because there's a harmony that lives there. All right, so continuing on. Given the definition of the partial order that we have, we wanna show that the symbol X join Y is the least upper bound with respect to this order. That is to say, to show that it's the least upper bound, we have to show that X and Y are less than X, X join Y. That's not too hard to see here because if I take X join X Y, because it's associative, you end up with X join X for which, since it's idempotent, you get X join Y. So this would give us that X is less than or equal to X join Y. And similarly, you're gonna get Y is less than or equal to X join Y. You might have to use commutivity in there, not a big deal, but all the accents come to play. So this tells us that X join Y is an upper bound of X and Y for which if you slightly define the partial order like we were doing here, this would actually give you a lower bound, but we'll stick with what we have here. Y is at the least upper bound. We'll take any other element, Z, in that situation. Suppose that X and Y are both less than or equal to Z, right? So that means that X join Z is equal to Z and Y join Z is equal to Z. So what about X join Y? Well, if you join that with Z, by associativity, this will be X join Y Z, but by assumption, Y Z is just a Z, and then by assumption, again, X Z is equal to Z in that situation. So this shows us that X join Z is less than or equal to Z and since Z was an upper bound and X join Y is less than every upper bound, that makes it the least upper bound. And so we see in this situation that given a semi lattice, you can put onto it a partially ordered set for which you always have least upper bounds. The least upper bound is in fact the quote unquote product of these things, the join of these things. Now, it turns out that this process is reversible. That is, if you start off with a set, if you start off with a set X and you have some partial order on it, such that least upper bounds always exist with respect to this partial order, then it turns out you can actually create a semi lattice by reversing this process. Since the least upper bound exists, you say that X join Y is equal to their least upper bound. You can then argue that this operation is associative, it's commutative and it is idempotent, thus giving you a semi lattice. Although I'm gonna leave that as an exercise to the viewer here.