Mathematical Structures: History of Commutative residuated partially ordered semigroups

# History of Commutative residuated partially ordered semigroups

 Revision 2 . . July 26, 2004 3:51 pm by Jipsen Revision 1 . . July 25, 2004 12:05 pm by Jipsen

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

Changed: 28,29c28,29
 \Large Commutative residuated partially ordered semigroups \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_residuated_partially_ordered_semigroups}{edit}

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

Changed: 34,37c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 A commutative residuated partially ordered semigroup is a \href{Residuated_partially_ordered_semigroups.pdf}{residuated partially ordered semigroup} $\mathbf{A}=\langle A, \cdot, \to, \le\rangle$ such that

Changed: 39,41c36
 $op_1$ is (name of property): $axiom_1$ $op_2$ is ...: $...$
 $\cdot$ is commutative: $xy=yx$

Changed: 50,51c45,47
 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 commutative residuated partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \to y)=h(x) \to h(y)$, and $x\le y\implies h(x)\le h(y)$.

Changed: 77c73
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & quasivariety \\\hline

Changed: 119,121c115
 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Commutative_residuated_lattice-ordered_semigroups.pdf}{Commutative residuated lattice-ordered semigroups} expanded type

Changed: 127c121
 \href{....pdf}{...} supervariety
 \href{Residuated_partially_ordered_semigroups.pdf}{Residuated partially ordered semigroups} same type

Changed: 129c123
 \href{....pdf}{...} subreduct
 \href{Commutative_partially_ordered_semigroups.pdf}{Commutative partially ordered semigroups} reduced type