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Note on axioms of a dually residuated lattice ordered semigroup. (English) Zbl 0906.06012

A dually residuated lattice-ordered semigroup was defined by K. L. N. Swamy [Math. Ann. 159, 105-114 (1965; Zbl 0135.04203)] as an algebraic system \((S;0,+,-, \wedge,\vee)\) satisfying the following axioms:
(1) \((S;0,+)\) is a commutative monoid,
(2) \((S;\wedge,\vee)\) is a lattice,
(3) \((a\wedge b)+ c=(a+c) \wedge(b+c)\),
(4) \((a\vee b) +c= (a+c) \vee(b+c)\),
(5) \(x+a\geq b\) iff \(x\geq b-a\),
(6) \(((a-b) \vee 0)+b\leq a\vee b\).
In this short note it is shown that axiom (3) follows from the others.
Remark. More generally, for any partially ordered groupoid which is a meet semilattice with respect to its partial order, axiom (3) follows from axiom (5) – see L. Fuchs, Partially ordered algebraic systems (1963; Zbl 0137.02001), the dual of Lemma A on page 190.
Reviewer: H.Mitsch (Wien)

MSC:

06F05 Ordered semigroups and monoids
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