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Direct product decompositions of bounded commutative residuated -monoids

Ján Jakubík (2008)

Czechoslovak Mathematical Journal

The notion of bounded commutative residuated -monoid ( B C R -monoid, in short) generalizes both the notions of M V -algebra and of B L -algebra. Let A ̧ be a B C R -monoid; we denote by ( A ̧ ) the underlying lattice of A ̧ . In the present paper we show that each direct...

Direct summands and retract mappings of generalized M V -algebras

Ján Jakubík (2008)

Czechoslovak Mathematical Journal

In the present paper we deal with generalized M V -algebras ( G M V -algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, G M V -algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of G M V -algebras. The relations between G M V -algebras and lattice ordered groups are essential for this investigation.

Directoids with sectionally antitone involutions and skew MV-algebras

Ivan Chajda, Miroslav Kolařík (2007)

Mathematica Bohemica

It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.

Distributivity of bounded lattices with sectionally antitone involutions

Ivan Chajda (2005)

Discussiones Mathematicae - General Algebra and Applications

We present a simple condition under which a bounded lattice L with sectionally antitone involutions becomes an MV-algebra. In thiscase, L is distributive. However, we get a criterion characterizingdistributivity of L in terms of antitone involutions only.

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