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Weak isometries and direct decompositions of dually residuated lattice ordered semigroups. (English) Zbl 0782.06012
A commutative lattice-ordered semigroup $$(S,+,\land,\lor,0)$$ with zero is called dually residuated (DRl-semigroup) if 1) for any $$a,b\in S$$ there exists a least $$x\in S$$, $$x=a-b$$, such that $$a\leq b+x$$, 2) $$(a- b)\lor 0+ b\leq a\lor b$$ for all $$a,b\in S$$, 3) $$a-a\geq 0$$ for each $$a\in S$$. A mapping $$f: S\to S$$ is called a weak 0-isometry in $$S$$ if $$f(0)=0$$ and $$d(x,y)= d(f(x),f(y))$$ for all $$x,y\in S$$, where $$d(a,b)= (a- b)\lor (b- a)$$ for $$a,b\in S$$. The relation between isometries and direct product decompositions in lattice-ordered groups, studied by Ch. Holland, J. Jakubik, W. Powell and the author, is shown to hold also for DRl- semigroups. Here, a commutative partially ordered semigroup $$(H,+,\leq)$$ with zero is said to be the direct product of its subsemigroups $$P$$ and $$Q$$, if 1) for each $$c\in H$$, $$c=c_ 1+ c_ 2$$ with unique $$c_ 1\in P$$, $$c_ 2\in Q$$, 2) if $$c=c_ 1+ c_ 2$$, $$d=d_ 1+ d_ 2$$ $$(c_ 1,d_ 1\in P, c_ 2,d_ 2\in Q)$$ then $$c\leq d$$ iff $$c_ 1\leq d_ 1$$ and $$c_ 2\leq d_ 2$$. The main result of the paper states that for any DRl-semigroup $$S$$ each weak 0-isometry $$f$$ defines a direct product decomposition of $$S$$ into the DRl-semigroup $$A=\{x\in S\mid f(x)=x\}$$ and the lattice-ordered group $$B=\{x\in S\mid f(x)=0-x\}$$.
Reviewer: H.Mitsch (Wien)

MSC:
 06F05 Ordered semigroups and monoids
Full Text:
References:
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