Normal autometrized lattice ordered algebras
Tomáš Kovář · Czech digital mathematics library · 2000
Results proved for normal autometrized lattice ordered algebras under the assumption of semiregularity are shown to be valid without this as sumption.Autometrized algebras were introduced by Swamy (cf.[6]) as an attempt to obtain a unified theory of abelian lattice ordered groups and Brouwerian algebras.Swamy and Rao (cf.[7]) studied the concept of an autometrized lattice ordered algebra.Swamy and Rao (cf.[7]) remarked that the notion of an autometrized algebra is too general and they introduced the notions of a normal autometrized algebra and a semiregular autometrized algebra.This work was continued by Hansen (cf.[1] and [2]) and Rachunek (cf.[3], [4] and [5]).In this paper we show that several results which were proved in the above quoted papers under the assumption of semiregularity can be proved without this assumption.We also give a characterization of an ideal of a normal autometrized lattice ordered algebra.An algebra A = (A; 0; +; A; V; *) of type (0;2;2;2;2) is a normal au tometrized lattice ordered algebra (abbreviated, NA^-algebra) if the following holds (cf.[6; Definition 1] and [7; Definition 1]): (i) (A; 0; +; <) is an abelian lattice ordered monoid, i.e.(a) (^4; 0; +) is an abelian monoid, (b) (A; A; V) is a lattice (the induced order is denoted by <), (c) x + (y A z) = (x + y) A (x + z) for all x,y,z e A, (d) x + (y V z) = (x + y) V (x + z) for all x, H, z G A, 2000 Mathematics Subject Classification: Primary 06F05.