Semi-Boolean lattices.

Nemitz, William C.

doi:10.1305/ndjfl/1093893707

Cited by 12 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The algebras in GH are term-equivalent to relative Stone algebras [27] or semi-Boolean lattices [37]; their properties have been thoroughly investigated. In particular, it is well-known that this variety and the variety of Heyting algebras generated by chains are locally finite and so the quasiequational theories of GH and GA are decidable.…”

Section: G-algebras and Their Reductsmentioning

confidence: 99%

Basic Hoops: an Algebraic Study of Continuous t-norms

Aglianò

Ferreirim²,

Montagna

2007

Stud Logica

View full text Add to dashboard Cite

A continuous t-norm is a continuous map * from [0, 1] 2 into [0, 1] such that [0, 1], * , 1 is a commutative totally ordered monoid. Since the natural ordering on [0, 1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and [0, 1], * , →, 1 becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras [0, 1], * , →, 1 , where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [0, 1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL-algebras, of Gödel algebras and of product algebras.A continuous t-norm is a continuous map * from [0, 1] 2 into [0, 1] such that [0, 1], * , 1 is a commutative totally ordered monoid. There are three fundamental continuous t-norms: the Lukasiewicz t-norm defined by x * L y = max(x + y − 1, 0), the Gödel (or lattice) norm x * G y = x ∧ y and the product norm x * P y = xy. Indeed it is known ([24, 35]) that, up to isomorphism, every continuous t-norm behaves locally as one of the above.Since the natural ordering on [0, 1] is a complete lattice ordering, each t-norm induces naturally a residuation, or an implication in more logical terms, by x → y = sup{z : z * x ≤ y}. The implications associated to the three fundamental norms are:x → L y = min(y − x + 1, 1)x → G y = 1 if x ≤ y y otherwiseResearch partly supported by research projects Praxis 2/ 2.

show abstract

Section: G-algebras and Their Reductsmentioning

confidence: 99%

Basic Hoops: an Algebraic Study of Continuous t-norms

Aglianò

Ferreirim²,

Montagna

2007

Stud Logica

View full text Add to dashboard Cite

show abstract

“…Concerning implication semilattices, i.e. join-semilattices where every section is a Boolean algebra, and concerning the structure and properties of such algebras, the reader is referred to [15], [16] and [17].…”

Section: Introductionmentioning

confidence: 99%

Join-semilattices whose principal filters are pseudocomplemented lattices

Chajda¹,

Länger²

2021

Preprint

View full text Add to dashboard Cite

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a∨ b in the section [b, 1] is denoted by a → b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with so-called non-classical implication semilattices which can be converted into I-algebras having everywhere defined operations. Moreover, we relate our structures to sectionally and relatively residuated semilattices which means that our logical structures are closely connected with substructural logics. We show that I-algebras form a congruence distributive, 3-permutable and weakly regular variety.

show abstract

“…There is another way to see that: As known, e.g., from [34], congruence relations of a Brouwerian semilattice are in 1-1 correspondence with filters. To be more precise: If F is a filter of a Brouwerian semilattice A then the relation 9F with (1.13) a9Fb<^a*b f\b * aG F is a congruence relation of A.…”

mentioning

confidence: 99%

“…In [34] he initiated an investigation of Brouwerian semilattices from a purely algebraic point of view. This has been continued by him and others, and more and more the influence of Universal Algebra as a rapidly developing research area became apparent as witnessed by the papers [40], [41] and-under a somewhat different aspect- [21].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Brouwerian semilattices

Köhler¹

1981

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let P be the category whose objects are posets and whose morphisms are partial mappings a: P-> Q satisfying (i) V p,q e dom a [p < q =* a(p) < a(q)] and The full subcategory Py of P consisting of all finite posets is shown to be dually equivalent to the category of finite Brouwerian semilattices and homomorphisms. Under this duality a finite Brouwerian semilattice A corresponds with M(A), the poset of all meet-irreducible elements of A. The product (in P,) of n copies (n E N) of a one-element poset is constructed; in view of the duality this product is isomorphic to the poset of meet-irreducible elements of the free Brouwerian semilattice on n generators.If V is a variety of Brouwerian semilattices and if A is a Brouwerian semilattice, then A is V-critical if all proper subalgebras of A belong to V but not A. It is shown that a variety V of Brouwerian semilattices has a finite equational base if and only if there are up to isomorphism only finitely many V-critical Brouwerian semilattices. This is used to show that a variety generated by a finite Brouwerian semilattice as well as the join of two finitely based varieties is finitely based. A new example of a variety without a finite equational base is exhibited. 0. Introduction. The interest in Brouwerian semilattices has been motivated mainly from two seemingly different directions. First there is intuitionistic propositional logic: The Lindenbaum algebra of its fragment £/c, which consists of formulas containing only implication and conjunction as logical connectives, is a free Brouwerian semilattice on a countable number of generators. Thus a formula is a tautology relative to £/c if and only if it is valid in every Brouwerian semilattice. In addition the validity relation between formulas and Brouwerian semilattices sets up a Galois connection between extensions of £/c and subvarieties of the variety of Brouwerian semilattices. Consequently algebraic methods can be successfully employed to deal with problems of a logical nature. Examples of this approach are the papers [31] and [52]; it must be said, however, that historically it was mostly full intuitionistic propositional logic (with disjunction and possibly negation) that was studied and thus on the algebraic side more interest was created in Brouwerian lattices and Heyting algebras.On the other hand the main pioneer in the study of Brouwerian semilattices, Nemitz, considered them as algebraic objects in their own right. In [34] he initiated

show abstract

Semi-Boolean lattices.

Cited by 12 publications

References 0 publications

Basic Hoops: an Algebraic Study of Continuous t-norms

Basic Hoops: an Algebraic Study of Continuous t-norms

Join-semilattices whose principal filters are pseudocomplemented lattices

Brouwerian semilattices

Contact Info

Product

Resources

About