Structure theorems for idempotent residuated lattices

Abstract

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery's characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…Despite the extensive literature devoted to classes of residuated lattices, there are still few effective structural descriptions. All these results in [1,12,22,23,25,33,34,36,42,46,50] (and also others where the focus is not a structural description in the first place such as in [20,21] for example) postulate semilinearity which renders the subdirectly irreducible members linearly ordered, and either integrality together with the naturally ordered condition 2 or idempotency. Our study contributes to the structural description of residuated lattices which are semilinear but neither integral nor naturally ordered nor idempotent; a first such effective structural description to the best of our knowledge besides [31,32].…”

Semilinear idempotent distributive l-monoids

“…The closed formula for I(n) is the same formula as in [21] for the number of idempotent residuated chains with n elements, but in [21] the number of n-element idempotent residuated chains should be I(n − 1), since every finite idempotent residuated chain is of the form C 2 M, where M is an idempotent ordered monoid, i.e., the number of idempotent residuated chains with n + 1 elements is equal to the number of idempotent ordered monoids with n elements. So an alternative proof of Corollary 3.8 can be obtained by using the result of [21] about idempotent residuated chain. Theorem 3.10.…”

Amalgamation in classes of involutive commutative residuated lattices

“…In residuated lattices the amalgamation property has been analyzed mostly in varieties in which the algebras are linear or semilinear, i.e., subdirect products of linearly ordered ones, or conic or semiconic. In addition, the investigated classes in the literature have mostly been either divisible and integral [10] or idempotent [1,4]. The scope of the present paper is investigating the amalgamation property in some varieties of residuated lattices which are neither divisible nor integral nor idempotent.…”

“…Also, the presence or failure of amalgamation is established for some subvarieties of residuated lattices, notably for all subvarieties of commutative GMV-algebras. As shown in [4], the variety generated by totally ordered commutative idempotent residuated lattices 2010 Mathematics Subject Classification. Primary 97H50, 20M30; Secondary 06F05, 06F20, 03B47.…”