The existence of lateral completions of `-groups is an old problem that was first solved, for conditionally complete vector lattices, by Nakano [5]. The existence and uniqueness of lateral completions of representable `-groups was first obtained as a consequence of the orthocompletions of Bernau [1], and later the proofs were simplified by Conrad [3], who also proved the existence and uniquenes...

This paper presents basic notions about fuzzy measures over algebras of fuzzy subsets of a fuzzy set. It also presents basic ideas on fuzzy integrals defined using these fuzzy measures. Definitions of new types of fuzzy measures and integrals are motivated by our research on generalized quantifiers. Several useful properties of fuzzy measures and fuzzy integrals are stated and proved. Definitio...

In 1998, Hájek established a representation theorem of BL-algebrs as all subdirect products of linear BL-algebrs. We establish a similar result for the much wider class of prelinear residuated algebras, in which neither the lattice structure nor the divisibility of the monoid operation is assumed. We show, in the case of prelinear residuated lattices, that this order embedding becomes a lattice...

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lat...

A commutative residuated lattice is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It proved here that epimorphisms surjective in variety K such algebras (with or without involution), provided each finitely subdirectly irreducible B has two properties: (1) generated by e, and (2) poset prime...

At first, we recall the basic concept, By a residual lattice is meant an algebra ) 1 , 0 , , , , , ( o ∗ ∧ ∨ = L L such that (i) ) 1 , 0 , , , ( ∧ ∨ = L L is a bounded lattice, (ii) ) 1 , , ( ∗ = L L is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: y z y x = ∗ ∨ ) ( if and only if y x z y o ≤ ≤ Let us note [7] that y x ∨ is the greatest element of the set y z y x ...

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follows, including various types of completeness theorems of substructural logics.

It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of posit...

Continuing with our general study of algebraic hyperstructures, we focus on the residuated operation in the framework of multilattices. Firstly, we recall the existing relation between filters, homomorphisms and congruences in the framework of multilattices; then, introduce the notion of residuated multilattice and further study the notion of filter, which has to be suitably modified so that th...