Generalized states on <i>EQ-</i>algebras

Abstract

In this paper, we introduce a notion of generalized states from an EQ-algebra epsilon(1) to another EQ-algebra epsilon(2), which is a generalization of internal states (or state operators) on an EQ-algebra epsilon. Also we give a type of special generalized state from an EQ-algebra epsilon(1 )to epsilon(1), called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) states on EQ-algebras. Moreover we discuss the relations between generalized states on EQ-algebras and internal states on other algebras, respectively. We obtain the following results: (1) Every state-morphism on a good EQ-algebra epsilon is a G-state from epsilon to the EQ-algebra epsilon(0) = ([0, 1](, )boolean AND(0), circle dot(0), similar to(0), 1). (2) Every state operator mu satisfying mu(x) circle dot mu(y) is an element of mu(epsilon) on a good EQ-algebra epsilon is a GI-state on epsilon. (3) Every state operator tau on a residuated lattice (L, boolean AND, V, circle dot, -> 0, 1) can be seen a GI-state on the EQ-algebra (L, boolean AND, circle dot, similar to, 1), where x similar to y := (x -> y) boolean AND (y -> x). (4) Every GI-state sigma on a good EQ-algebra (L, boolean AND, circle dot, similar to, 1) is a internal state on equality algebra (L, boolean AND, similar to, 1). (5) Every GI-state sigma on a good EQ-algebra (L, boolean AND, circle dot, similar to, 1), is a left state operator on BCK-algebra (L, boolean AND, ->, 1), where x -> y = x similar to x boolean AND y.