Kovář, Tomáš Note on axioms of a dually residuated lattice ordered semigroup. (English) Zbl 0906.06012 Discuss. Math., Algebra Stoch. Methods 17, No. 1, 89-90 (1997). 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) Cited in 1 Document MSC: 06F05 Ordered semigroups and monoids Keywords:dually residuated lattice-ordered semigroup Citations:Zbl 0135.04203; Zbl 0137.02001 PDFBibTeX XMLCite \textit{T. Kovář}, Discuss. Math., Algebra Stoch. Methods 17, No. 1, 89--90 (1997; Zbl 0906.06012)