Extremal states on bounded residuated $$\ell$$ -monoids with general comparability

Abstract

Bounded residuated lattice ordered monoids ($$R\ell$$-monoids) are a common generalization of pseudo-$$BL$$-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. We investigate bounded $$R\ell$$-monoids satisfying the general comparability condition in connection with their states (analogues of probability measures). It is shown that if an extremal state on Boolean elements fulfils a simple condition, then it can be uniquely extended to an extremal state on the $$R\ell$$-monoid, and that if every extremal state satisfies this condition, then the $$R\ell$$-monoid is a pseudo-$$BL$$-algebra.