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Varieties of commutative residuated integral pomonoids and their residuation subreducts. (English) Zbl 0872.06007
Let \(\langle A;\oplus ,0,\leq \rangle\) be a commutative (dually) integral partially ordered monoid whose identity \(0\) is the least element of \(\langle A,\leq \rangle\), where \(\leq\) is a partial order compatible with the monoid operation \(\oplus\) in the sense that \(a\oplus b\leq c\oplus d\) whenever \(a\leq c\) and \(b\leq d\). If for each \(a,b\in A\) there is a least element \(c\in A\) (denoted \(a\div b\)) such that \(a\leq c\oplus b\), then the resulting structure \(\langle A;\oplus ,\div ,0,\leq \rangle\) is called a partially ordered commutative (dually) residuated (dually) integral monoid (briefly a pocrim). The operation \( \div \) is called residuation. BCK-algebras are the residuation subreducts of pocrims. The simplest example of a pocrim is the set of ideals of a commutative ring with \(1\) with respect to the ideal multiplication as the monoid operation and the (lattice) order of reversed set inclusion. Residuation is defined by \( I\div J=I:J\).
The class of all pocrims is a quasivariety \( \mathcal{V} \) which is not a variety, but it is relatively congruence distributive, has the relative congruence extension property, and is relatively point regular with respect to the monoid identity \(0\). All \(0\)-regular subvarieties are congruence \(n\)-permutable for some \(n\). In several varieties of pocrims (for example in the variety of Brouwerian semilattices and the variety of hoops) the finitely generated subdirectly irreducible algebras can be obtained from a finite number of simple algebras in the variety by a finite number of applications of the operations of variety generation and ordinal sum. Cancellative pocrims form a subquasivariety \( \mathcal{C} \) which has a Mal’cev term and generates a congruence permutable variety (not contained in \( \mathcal{V} \)). The subvarieties of \( \mathcal{C} \) form a lattice which has no greatest element.

MSC:
06B20 Varieties of lattices
13A15 Ideals and multiplicative ideal theory in commutative rings
06F05 Ordered semigroups and monoids
06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
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