Monadic Wajsberg hoops

Abstract

Wajsberg hoops are the { , →, 1}-subreducts (hoop-subreducts) of Wajsberg algebras, which are term equivalent to MV-algebras and are the algebraic models of Lukasiewicz infinite-valued logic. Monadic MV-algebras were introduced by Rutledge [Ph.D. thesis, Cornell University, 1959] as an algebraic model for the monadic predicate calculus of Lukasiewicz infinitevalued logic, in which only a single individual variable occurs. In this paper we study the class of { , →, ∀, 1}-subreducts (monadic hoop-subreducts) of monadic MV-algebras. We prove that this class, denoted by MWH, is an equational class and we give the identities that define it. An algebra in MWH is called a monadic Wajsberg hoop. We characterize the subdirectly irreducible members in MWH and the congruences by monadic filters. We prove that MWH is generated by its finite members. Then, we introduce the notion of width of a monadic Wajsberg hoop and study some of the subvarieties of monadic Wajsberg hoops of finite width k. Finally, we describe a monadic Wajsberg hoop as a monadic maximal filter within a certain monadic MValgebra such that the quotient is the two element chain.