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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
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