DRl-monoids. (English) Zbl 0828.06009

Chajda, I. (ed.) et al., General algebra and ordered sets. Proceedings of the international conference and summer school, held in Horní Lipová, Czech Republic, September 4-12, 1994. Olomouc: Palacký University Olomouc, Department of Algebra and Geometry, 79-83 (1994).
Dually residuated lattice-ordered (DRl-)monoids were introduced and studied by K. L. N. Swamy in the commutative case as a common abstraction of abelian lattice-ordered groups and Brouwerian (Boolean) algebras [Math. Ann. 159, 105-114 (1965; Zbl 0135.042), ibid. 160, 64-71 (1965; Zbl 0138.021) and ibid. 167 , 71-74 (1966; Zbl 0158.026)]. In the paper under review, noncommutative DRl-monoids are defined adding two further axioms which respect left and right residuation. Again, lattice- ordered groups and Brouwerian algebras form examples. Some properties of these general DRl-monoids, similar to those obtained by K. L. Swamy (loc. cit.), are derived – but proofs are not given in general.
Remark: In Theorem 6, the condition: \(x + y = y + x\) for all \(x \in \text{Inv} A\) and all \(y \in \text{Sing} A\), should be added.
For the entire collection see [Zbl 0815.00007].
Reviewer: H.Mitsch (Wien)


06F05 Ordered semigroups and monoids