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Direct decompositions of dually residuated lattice ordered monoids. (English) Zbl 1068.06016
Dually residuated lattice-ordered monoids $$(M,+,0,\vee,\wedge)$$ (DRl-monoids) were defined by T. Kovář by the usual axioms adding four new ones (compare with G. Birkhoff or L. Fuchs). This concept is a common generalization of $$l$$-groups, Brouwerian lattices, MV- and GMV-algebras, BL-and pseudo BL-algebras. An ideal of a DRl-monoid $$M$$ is a non-empty subset $$I$$ of $$M$$ satisfying:
(i) $$x,y\in I\Rightarrow x+ y\in I$$, and
(ii) $$x\in I$$, $$y\in M$$, $$|y|\leq|x|\Rightarrow y\in I$$.
In this paper, direct products of DRl-monoids are studied. For example, it is shown that, for ideals $$I$$, $$J$$ of $$M$$ such that $$I+ J= M$$, $$I\cap J= \{0\}$$ and $$x+ y= x'+y'$$ $$(x,x'\in I,y,y'\in J)\Rightarrow x= x'$$, $$y= y'$$, $$M$$ is isomorphic with the direct product $$I\times J$$. Also, if $$M$$ satisfies these conditions except possibly the first, $$I+ J= M$$, then the direct factors in $$M$$ form a Boolean sublattice of the lattice of all ideals in $$M$$.
Reviewer: H. Mitsch (Wien)

MSC:
 06F05 Ordered semigroups and monoids
Zbl 0828.06009
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