Quasi-varieties of metric algebras
An \(M\)-algebra or a metric algebra is an abstract algebra endowed with a metric which assumes values in \([0, \infty]\). In a natural way the author introduces such notions as \(M\)-isomorphism, \(M\)-subalgebra, reduced \(M\)- product, and he defines operators \(I\), \(S\), \(P^ M_ R\) corresponding to the above operations. An \(M\)-quasivariety is a class of \(M\)-algebras closed under \(I\), \(S\), \(P_ R^ M\). The key notion of this paper is that of equicontinuous satisfaction (it is an analogue of uniform continuity in the usual sense). The author proves that a class of metric algebras is an \(M\)-quasivariety iff it is definable by finite implications and equicontinuously satisfies every finite implication that it satisfies. It is shown that an \(M\)- quasivariety obeys a subdirect representation theorem and admits completion of its members. Also the author finds \(K\)-irreducible members without infinite distances in the class \(K\) of real metric vector spaces.
scientific article; zbMATH DE number 724098