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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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