# Differences

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distributive_lattice_ordered_semigroups [2010/07/29 15:46]
127.0.0.1 external edit
distributive_lattice_ordered_semigroups [2018/10/14 16:16] (current)
jipsen
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$\cdot$ distributes over $\vee$:  $x\cdot(y\vee z)=(x\cdot y)\vee (x\cdot z)$ and $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z)$ $\cdot$ distributes over $\vee$:  $x\cdot(y\vee z)=(x\cdot y)\vee (x\cdot z)$ and $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z)$
-
-Remark: This is a template.
-
-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== ==Morphisms==
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$h(x\cdot y)=h(x) \cdot h(y)$ $h(x\cdot y)=h(x) \cdot h(y)$
-====Definition==== +====Examples====
-An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +Example 1: Any collection$\mathbf A$of binary relations on a set$X$such that$\mathbf A$is closed under union, intersection and composition. -...\rangle$ such that+
-$...$ is ...:  $axiom$ +H. Andreka[(Andreka1991)] proves that these examples generate the variety DLOS.
-   +
-$...$ is ...:  $axiom$ +
- +
-====Examples==== +
-Example 1: +
====Basic results==== ====Basic results====
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$\begin{array}{lr}$\begin{array}{lr}
f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &6\\
-  f(3)= &\\ +  f(3)= &44\\
-  f(4)= &\\+  f(4)= &479\\
f(5)= &\\   f(5)= &\\
-\end{array}$-$\begin{array}{lr}
-  f(6)= &\\
-  f(7)= &\\
-  f(8)= &\\
-  f(9)= &\\
-  f(10)= &\\
\end{array}$\end{array}$
-
====Subclasses==== ====Subclasses====
-  [[...]] subvariety +[[Distributive lattice-ordered monoids]]
- +
-  [[...]] expansion+
+[[Commutative distributive lattice-ordered semigroups]]
====Superclasses==== ====Superclasses====
-  [[...]] supervariety +[[Lattice-ordered semigroups]]
- +
-  [[...]] subreduct +
====References==== ====References====
-[(Andreka1991> +[(Andreka1991>Hajnal Andreka, \emph{Representations of distributive lattice-ordered semigroups with binary relations}, Algebra Universalis \textbf{28} (1991), 12--25)]
-Hajnal Andr\'eka, \emph{Representations of distributive lattice-ordered semigroups with binary relations}, Algebra Universalis \textbf{28} (1991), 12--25  +
-[[MRreview]] +
-)]+

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