Direct decompositions of dually residuated lattice ordered monoids. (English) Zbl 1068.06016

Rachůnek, Jiří; Šalounová, Dana

Discuss. Math., Gen. Algebra Appl. 24, No. 1, 63-74 (2004).

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

Keywords:

Dually residuated lattice-ordered monoids; direct products; ideals

Citations:

Zbl 0828.06009