User:Jshflynn

A Relation Zoo
What follows is a table of specific types of transitive relations as they are named on.

The relation of interest shall be denoted $\mathcal R$ and it shall be on the set $S$.

Under Reflexivity

 * T means the relation is reflexive.


 * F means the relation is antireflexive.

These can both hold iff $S = \varnothing$.

Under Symmetry

 * T means the relation is symmetric.


 * F means the relation is antisymmetric.

These can both hold iff $\mathcal R$ is coreflexive as proved in Relation both Symmetric and Antisymmetric.

If they both hold and $\mathcal R$ is reflexive or connected then $\mathcal R$ is the diagonal relation on $S$.

If they both hold and $\mathcal R$ is antireflexive then $\mathcal R$ is the null relation on $S$.

We leave it blank if it is not of concern to us.

Under Connected

 * T means the relation is connected.


 * F means it is not.

We leave it blank if it is not of concern to us.

The Table
In addition to this we name:


 * Partial Preordering: A Weak Partial Preordering or Strict Partial Preordering.


 * Total Preordering: A Weak Total Preordering or Strict Total Preordering.


 * Weak Preordering: A Weak Partial Preordering or Weak Total Preordering.


 * Strict Preordering: A Strict Partial Preordering or Strict Total Preordering.


 * Preordering: A Weak Preordering or Strict Preordering.

Similarly we name:


 * Partial Ordering: A Weak Partial Ordering or Strict Partial Ordering.


 * Total Ordering: A Weak Total Ordering or Strict Total Ordering.


 * Weak Ordering: A Weak Partial Ordering or Weak Total Ordering.


 * Strict Ordering: A Strict Partial Ordering or Strict Total Ordering.


 * Ordering: A Weak Ordering or Strict Ordering.

For the relational structure $\left({S, \mathcal R}\right)$ we name it as follows:

Blah blah blah :) --Jshflynn (talk) 13:06, 14 March 2013 (UTC)

True or not?
A non-empty semigroup $(S, \circ)$ is a group iff:


 * $\forall x \in S: x \circ_\mathcal P S = S \circ_\mathcal P x = S$

This definition of a group is given in An Introduction to Semigroup Theory (by John Howie). He states that its convenient for his set up but acknowledges that it is a highly unusual definition to give for a group.

Lame
Something occurred earlier to do with my cookies. My proof was not saved :(

--Jshflynn (talk) 15:13, 9 March 2013 (UTC)


 * If you use Google Chrome there exists the ability to go back to the page that failed to be saved and you can then do an emergency cut-and-paste into a separate text editor from which you can then retrieve the material you would otherwise lose. --prime mover (talk) 16:01, 9 March 2013 (UTC)


 * I did not know that. Thanks :) --Jshflynn (talk) 16:12, 9 March 2013 (UTC)

Sandbox
User:Jshflynn/Sandbox0

User:Jshflynn/Sandbox1

User:Jshflynn/Sandbox2

User:Jshflynn/Left Zero Semigroup

User:Jshflynn/Right Zero Semigroup

User:Jshflynn/Rectangular Band Isomorphism Theorem

Equivalence Relation Alternative Definition 2
$(1)$ $\mathcal R$ is left total or right total.

$(2)$ $\mathcal R = \mathcal R^{-1}$

$(3)$ $\mathcal R \circ \mathcal R = \mathcal R$

Construct Notes
Presenting Relations:

$(a \mathrel{\mathcal R} b)$

$(a \mathcal R b)$

As for parentheses, you should be careful because $\lor$, $\land$, and $\lnot$ can have somewhat different meanings that can be confused without parentheses. For example, $a = b \wedge c = d$ could be read as $(a = b) \land (c = d)$ (that is, "$a = b$ and $c = d$", or it could be read as "$a = (b \wedge c) = d$" (that is, "$a$ equals the meet of $b$ and $c$, which also equals $d$). The same problem happens with $\lor$, which looks just like $\vee$, which can mean "join". Note also that $\neg$ may be read as "complement" in some cases, so you have to be just as careful there. --Dfeuer (talk) 07:59, 5 March 2013 (UTC)

Engine Fuel
I will be extracting some stuff off this if you don't mind:




 * Actually, I'd rather see it coming from some elements from the extensive reference list &mdash; notes like these can perish without warning. If you like the presentation, please make sure to grab a copy of it before it's too late. &mdash; Lord_Farin (talk) 07:40, 5 March 2013 (UTC)


 * Okay :) --Jshflynn (talk) 07:50, 5 March 2013 (UTC)

When did transclusion first appear on this site?
In particular when did it really take off?

There's something oddly pleasing about it.


 * Sometime around when I discovered its use in Trigonometric Identities, around 29th Dec 2010. My playlist at the time was the Beatles 1962-66 and 1967-70 which I'd just bought for the wife for xmas. Can you think of a more pleasurable occupation than doing maths while listening to the Beatles? --prime mover (talk) 21:24, 6 March 2013 (UTC)

Archive
Welcome to the archive. Unfortunately there are no staff here so note down landmarks and try not to get lost :)

User:Jshflynn/archive/preMarch2013