Hyper Commutative Basic Algebras, Hyper MV-algebras, and States

摘要

In this paper, we introduce hyper commutative basic algebras. We first prove that every hyper MV-algebra is a hyper commutative basic algebra. Then we show that a hyper commutative basic algebra of cardinality 2 is a hyper MV-algebra, which is similar to the result of Botur and Halas that finite commutative basic algebras coincide with MV-algebras. However, we find a hyper commutative basic algebra of cardinality 3 which is not a hyper MV-algebra. Finally, we study two types of states on hyper commutative basic algebras.