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—		lattice-ordered_semigroups [2010/07/29 15:46] (current)
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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]]
		+	)]
		+
		+