Commutative equality algebras and &-equality algebras

Abstract

The notion of &-equality algebras is introduced, and related properties are investigated. Using &-equality algebras, (commutative ordered) semigroups are induced. Conditions for an equality algebra to be an &-equality algebra and commutative equality algebra are provided. The concept of terminal section of an element is introduced, and several properties are studied. Using the notion of terminal section, conditions for an equality algebra to be a commutative equality algebra are considered. Given a subset of an equality algebra, the ◦-set and the *-set are introduced, and then several related properties are displayed. Conditions for the ◦-set (resp., *-set) to be deductive systems are stated.