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.