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+ | =====Lattice-ordered semigroups===== | ||
+ | |||
+ | Abbreviation: **LSgrp** | ||
+ | |||
+ | ====Definition==== | ||
+ | A \emph{lattice-ordered semigroup} (or \emph{$\ell$-semigroup}) is a structure $\mathbf{A}=\langle A\vee,\wedge,\cdot\rangle$ of type $\langle 2,2,2\rangle$ such that | ||
+ | |||
+ | $\langle A,\vee,\wedge\rangle$ is a [[lattice]] | ||
+ | |||
+ | $\langle A,\cdot\rangle$ is a [[semigroups]] | ||
+ | |||
+ | $\cdot$ distributes over $\vee$: $x(y\vee z)=xy\vee xz$, $(x\vee y)z=xz\vee yz$ | ||
+ | |||
+ | 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 $\mathbf{A}$ and $\mathbf{B}$ be lattice-ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: | ||
+ | $h(x \vee y)=h(x) \vee h(y)$, | ||
+ | $h(x \wedge y)=h(x) \wedge h(y)$, | ||
+ | $h(x \cdot y)=h(x) \cdot h(y)$, | ||
+ | |||
+ | ====Definition==== | ||
+ | A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle | ||
+ | ...\rangle$ such that | ||
+ | |||
+ | $...$ is ...: $axiom$ | ||
+ | |||
+ | $...$ is ...: $axiom$ | ||
+ | |||
+ | ====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]] |variety | | ||
+ | ^[[Equational theory]] | | | ||
+ | ^[[Quasiequational theory]] | | | ||
+ | ^[[First-order theory]] | | | ||
+ | ^[[Locally finite]] | | | ||
+ | ^[[Residual size]] | | | ||
+ | ^[[Congruence distributive]] |yes | | ||
+ | ^[[Congruence modular]] |yes | | ||
+ | ^[[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==== | ||
+ | [[Lattice-ordered monoids]] expanded type | ||
+ | |||
+ | |||
+ | ====Superclasses==== | ||
+ | [[Semigroups]] reduced type | ||
+ | |||
+ | [[Lattices]] reduced type | ||
+ | |||
+ | |||
+ | ====References==== | ||
+ | |||
+ | [(Lastname19xx> | ||
+ | F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] | ||
+ | )] | ||
+ | |||
+ | |||
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