A non-commutative generalization of MV-algebras

Abstract

A non-commutative generalization of M V -algebras

Monadic GMV-algebras

“…Proof By [18], the variety GMV of all GM V -algebras (of type 2, 1, 1, 0 ) is arithmetical, hence the variety MGMV is arithmetical, too.…”

Regularity of Generalized Mv-Algebras

“…[12]) to arbitrary 1-groups to obtain algebras which axe essentially more general than MV-algebras, the so-called generalized MV-algebras. Independently an equivalent generalization was introduced by G. Georgescu and A. Iorgulescu (cf.…”

“…[12]). In this case the unary operations -i and ~ coincide and the type can be reduced by omitting one of these operations.…”

“…Examples of generalized MV-algebras which are not MV-algebras are given by intervals of non-commutative 1-groups. Indeed, let Q = (G ,+,-,0,V,A) be an arbitrary 1-group and 0 < u G G. Then by Theorem 20 of [12], T(g,u) = ([0,u],©,©,-i,~,0,u) is a generalized MV-algebra. Recently A.Dvurecenskij showed that every generalized MV-algebra can be represented in this form (cf.…”

“…For a generalized MV-algebra A -(A, ©, ©, 0,1) the derived operations V and A are lattice operations with least element 0 and greatest element 1 (see [12]). In [10] it is proved that this lattice is distributive and by [12] for all x, y € A it holds…”