# Differences

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+ | =====Equivalence relations===== | ||

+ | Abbreviation: **EqRel** | ||

+ | |||

+ | ====Definition==== | ||

+ | An \emph{equivalence relation} is a structure $\mathbf{X}=\langle X,\equiv\rangle$ such that $\equiv$ is a \emph{binary relation on $X$} | ||

+ | (i.e. $\equiv\ \subseteq X\times X$) that | ||

+ | is | ||

+ | |||

+ | reflexive: $x\equiv x$ | ||

+ | |||

+ | symmetric: $x\equiv y\Longrightarrow y\equiv x$ | ||

+ | |||

+ | transitive: $x\equiv y\text{ and }y\equiv z\Longrightarrow x\equiv z$ | ||

+ | |||

+ | Remark: This is a template. | ||

+ | If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page. | ||

+ | |||

+ | It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes. | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{X}$ and $\mathbf{Y}$ be equivalence relations. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:A\rightarrow B$ that is a homomorphism: | ||

+ | $x\equiv^{\mathbf X} y\Longrightarrow h(x)\equiv^{\mathbf Y}h(y)$ | ||

+ | |||

+ | ====Definition==== | ||

+ | An \emph{equivalence relation} is a [[qoset]] that is \emph{symmetric}: $x\equiv y\Longrightarrow y\equiv x$ | ||

+ | |||

+ | ====Examples==== | ||

+ | Example 1: | ||

+ | |||

+ | ====Basic results==== | ||

+ | Equivalence relations are in 1-1 correspondence with [[partitions]]. | ||

+ | |||

+ | |||

+ | ====Properties==== | ||

+ | Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. | ||

+ | |||

+ | ^[[Classtype]] |quasivariety | | ||

+ | ^[[Quasiequational theory]] | | | ||

+ | ^[[First-order theory]] | | | ||

+ | ^[[Locally finite]] |yes | | ||

+ | ^[[Residual size]] | | | ||

+ | ^[[Congruence distributive]] |no | | ||

+ | ^[[Congruence modular]] |no | | ||

+ | ^[[Congruence $n$-permutable]] | | | ||

+ | ^[[Congruence regular]] | | | ||

+ | ^[[Congruence uniform]] | | | ||

+ | ^[[Congruence extension property]] | | | ||

+ | ^[[Definable principal congruences]] | | | ||

+ | ^[[Equationally def. pr. cong.]] | | | ||

+ | ^[[Amalgamation property]] | | | ||

+ | ^[[Strong amalgamation property]] | | | ||

+ | ^[[Epimorphisms are surjective]] | | | ||

+ | |||

+ | ====Finite members==== | ||

+ | |||

+ | $\begin{array}{lr} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &2\\ | ||

+ | f(3)= &3\\ | ||

+ | f(4)= &5\\ | ||

+ | f(5)= &7\\ | ||

+ | \end{array}$ | ||

+ | $\begin{array}{lr} | ||

+ | f(6)= &11\\ | ||

+ | f(7)= &15\\ | ||

+ | f(8)= &22\\ | ||

+ | f(9)= &30\\ | ||

+ | f(10)= &42\\ | ||

+ | \end{array}$ | ||

+ | |||

+ | The number of (labelled) equivalance relations on an $n$ element set given by a sum of Stirlings formula (of the second kind). | ||

+ | |||

+ | see also http://www.research.att.com/projects/OEIS?Anum=A000110 | ||

+ | |||

+ | The number of (nonisomorphic) equivalence relations is the number of partition patterns (= number of integer partitions). | ||

+ | |||

+ | see also http://www.research.att.com/projects/OEIS?Anum=A000041 | ||

+ | |||

+ | ====Subclasses==== | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Preordered sets]] supervariety | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Lastname19xx> | ||

+ | F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] | ||

+ | )] |

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