Mathematical Structures: History of Lattice-ordered semigroups

# History of Lattice-ordered semigroups

 Revision 2 . . July 30, 2004 12:35 pm by Jipsen Revision 1 . . July 29, 2004 12:35 pm by Jipsen

Difference (from prior major revision) (no other diffs)

Changed: 28,29c28,29

Changed: 31c31
 \abbreviation{Abbr}
 \abbreviation{LSgrp}

Changed: 34,35c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that
 A lattice-ordered semigroup (or $\ell$-semigroup) is a structure $\mathbf{A}=\langle A\vee,\wedge,\cdot\rangle$ of type $\langle 2,2,2\rangle$ such that

Changed: 37c36
 $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 $\langle A,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{lattice}

Changed: 39c38
 $op_1$ is name of property: $axiom_1$
 $\langle A,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups}

Changed: 41c40
 $op_2$ is ...: $...$
 $\cdot$ distributes over $\vee$: $x(y\vee z)=xy\vee xz$, $(x\vee y)z=xz\vee yz$

Changed: 50,51c49,52
 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 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: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$,

Changed: 77c78
 Classtype & (value, see description) \cite{Lastname19xx} \\\hline
 Classtype & variety \\\hline

Changed: 83,84c84,85
 Congruence distributive & \\\hline Congruence modular & \\\hline
 Congruence distributive & yes\\\hline Congruence modular & yes\\\hline

Changed: 119,121c120
 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Lattice-ordered_monoids.pdf}{Lattice-ordered monoids} expanded type

Changed: 127c126
 \href{....pdf}{...} supervariety
 \href{Semigroups.pdf}{Semigroups} reduced type

Changed: 129c128
 \href{....pdf}{...} subreduct
 \href{Lattices.pdf}{Lattices} reduced type