Quasi-Decompositions, Exact Sequences, and Triple Sums of Semigroups. Ii. Applications

Schmidt, Jürgen

doi:10.1016/b978-0-7204-0725-9.50035-6

Cited by 7 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Schmidt [11] developed the triple method in his "Construction Theorem" using equivalence classes of ordered pairs (a, d), a ∈ C, d ∈ D. Following Theorem 3.8 ϑ(a) is always a congruence relation generated by collapsing a principal filter, and therefore, ϑ can simply be replaced by any 1-homomorphism C → D. Projective semilattices have been characterized by A. Horn and N. Kimura [5,Theorem 5.3]. They already achieved several simple characterizations of projective semilattices, which are easy applicable.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Projective extensions of semilattices

Katriňák

Guričan

2006

Algebra univers.

View full text Add to dashboard Cite

show abstract

Section: Discussionmentioning

confidence: 99%

“…Our next aim is to show how we can construct semilattices with identity from a suitable triple. We shall proceed as in [11]. The preceding discussion motivates the following: Definition 4.2.…”

Section: Vol 55 2006mentioning

confidence: 99%

Projective extensions of semilattices

Katriňák

Guričan

2006

Algebra univers.

View full text Add to dashboard Cite

show abstract

“…What is less known is the following reverse construction, which has its origin in the general theory of 'triple sums' originally developed in [4,5] and later extended in [7][8][9][10].…”

Section: Latticesmentioning

confidence: 99%

“…Let us note that the theorem above could have also formulated using the notion of split exact sequences (see [9,10]).…”

Section: Latticesmentioning

confidence: 99%

The Central Decomposition of FD01(n)

Köhler

2021

Order

View full text Add to dashboard Cite

show abstract

“…More recently total subalgebras occurred-under the name Brouwerian subacts-in the theory of quasi-decompositions of Brouwerian semilattices as developed by Schmidt [45], [46]. A detailed discussion of the lattice of total subalgebras of a Brouwerian semilattice can be found in [23].…”

mentioning

confidence: 99%

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

Quasi-Decompositions, Exact Sequences, and Triple Sums of Semigroups. Ii. Applications

Cited by 7 publications

References 0 publications

Projective extensions of semilattices

Projective extensions of semilattices

The Central Decomposition of FD01(n)

Brouwerian semilattices

Contact Info

Product

Resources

About