# Differences

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+ | =====Symmetric relations===== | ||

+ | Abbreviation: **SymRel** | ||

+ | |||

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

+ | A \emph{symmetric relation} is a structure $\mathbf{X}=\langle X,R\rangle$ such that $R$ is a \emph{binary relation on $X$} | ||

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

+ | is | ||

+ | |||

+ | symmetric: $xRy\Longrightarrow yRx$ | ||

+ | |||

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

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

+ | |||

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

+ | |||

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

+ | Example 1: | ||

+ | |||

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

+ | |||

+ | |||

+ | ====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)= &\\ | ||

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

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

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

+ | \end{array}$ | ||

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

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

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

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

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

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

+ | \end{array}$ | ||

+ | |||

+ | |||

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

+ | |||

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

+ | [[Directed graphs]] supervariety | ||

+ | |||

+ | |||

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

+ | |||

+ | [(Lastname19xx> | ||

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

+ | )] |

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