Pseudocomplemented Semilattices Research Articles

Semilattice Pseudo-complements on Semigroups#

The notion of a pseudo-complement is extended to a broad class of semigroups, which we say are semilattice pseudo-complemented (SP). We examine the relationship between these and some other classes of unary semigroups, namely semigroups with closure and interior operations, and we also consider the concepts in a ring theoretic setting. Under some natural strengthenings of the defining axioms, some fundamental congruences are described, extending cases from the theory of inverse semigroups. The class of SP-semilattices coincides with a natural class of ordered structures related to topological spaces but differs slightly from the standard definition of a pseudo-complemented semilattice. These SP-semilattices arise naturally in the general theory and are given particular attention.

Pseudocomplemented Semilattices, Boolean Algebras, and Compatible Products

Pseudocomplemented semilattices are studied here from an algebraic point of view, stressing the pivotal role played by the pseudocomplements and the relationship between pseudocomplemented semilattices and Boolean algebras. Following the pattern of semiprime ring theory, a notion of Goldie dimension is introduced for complete pseudocomplemented lattices and calculated in terms of maximal uniform elements if they exist in abundance. Products in lattices with 0-element are studied and questions about the existence and uniqueness of compatible products in pseudocomplemented lattices, as well as about the abundance of prime elements in lattices with a compatible product, are discussed. Finally, a Yood decomposition theorem for topological rings is extended to complete pseudocomplemented lattices.

Free Products of Pseudocomplemented Semilattices

Free Products of Pseudocomplemented Semilattices

Congruences on pseudocomplemented semilattices

It is known that congruence lattices of pseudocomplemented semilattices are pseudocomplemented [4]. Many interesting properties of congruences on pseudocomplemented semilattices were described by Sankappanavar in [4], [5], [6]. Except for other results he described congruence distributive pseudocomplemented semilattices [6] and he characterized pseudocomplemented semilattices whose congruence lattices are Stone, i.e. belong to the variety B1 [5]. In this paper we give a partial solution to a more general question: Under what condition on a pseudocomplemented semilattice its congruence lattice is element of the variety Bn (n ‚ 2)? In the last section we widen the Sankappanavar’s result to obtain the description of pseudocomplemented semilattices with relative Stone congruence lattices. A partial solution of the description of pseudocomplemented semilattices with relative (Ln)-congruence lattices (n ‚ 2) is also given.

Ring-like operations is pseudocomplemented semilattices

Ring-like operations is pseudocomplemented semilattices

Congruence Lattices of Pseudocomplemented Semilattices

Congruence Lattices of Pseudocomplemented Semilattices

Quasivarieties of pseudocomplemented semilattices

Two properties of the lattice of quasivarieties of pseudocomplemented semilattices are established, namely, in the quasivariety generated by the 3-element chain, there is a sublattice freely generated by ω elements and there are $2^ω$ quasivarieties.

Finite semilattices whose non-invertible endomorphisms are products of idempotents

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.

On amalgamation classes of pseudocomplemented semilattices

On amalgamation classes of pseudocomplemented semilattices

A Construction for Pseudocomplemented Semilattices and Two Applications

A method is given by which pseudocomplemented semilattices can be constructed from graphs. Two consequences of the method are obtained, namely: there exist continuum-many quasivarieties of pseudocomplemented semilattices; for any non-zero cardinal $\kappa$, there exist $\kappa$ pairwise non-isomorphic pseudocomplemented semilattices with isomorphic endomorphism monoids.

A construction for pseudocomplemented semilattices and two applications

A method is given by which pseudocomplemented semilattices can be constructed from graphs. Two consequences of the method are obtained, namely: there exist continuum-many quasivarieties of pseudocomplemented semilattices; for any non-zero cardinal κ \kappa , there exist κ \kappa pairwise non-isomorphic pseudocomplemented semilattices with isomorphic endomorphism monoids.

Lee classes and sentences for pseudocomplemented semilattices

Lee classes and sentences for pseudocomplemented semilattices

Undecidability of Free Pseudo-Complemented Semilattlces

Decision problem for the first order theory of free objects in equational classes of algebras was investigated for groups (Malcev [10]), semigroups (Quine [12]), commutative semigroups (Mostowski [11]), distributive lattices (Ershov [6]) and several varieties of rings (Lavrov [9]). Recently this question was solved for all varieties of Hilbert algebras and distributive pseudo-complemented lattices (see [7], [8]). In this paper we prove that the theory of all finitely generated free pseudo-complemented semilattices is undecidable.

Atoms, Primes and Implicative Lattices

AbstractLet L be an a-implicative semilattice. We obtain a characterization of those elements which cover a. This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L, the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.

Congruence-semimodular and congruence-distributive pseudocomplemented semilattices

Investigations into the structure of the congruence lattices of pseudocomplemented semilattices (PCS's) were initiated in [10]. In this paper a we characterize the class of congruence-semim odular PCS's (i.e. PCS's with semimodular lattice of congruences) and the class of congruence-distr ibutive PCS's (i.e. with distributive congruence lattices). We give two characterizations of each class; one of these is a Dedekind-Birkhof f-type characterization which says that the exclusion in a certain sense of a single PCS P6 determines the class of congruence-semim odular PCS's, and the exclusion of the two PCS's P6 and P5 (these are defined in the sequel) determines the class of congruence-distr ibutive PCS's. The other characterization shows that each of these classes is strictly elementary and gives explicitly the defining axiom for each class as a universal positive sentence (in the language of PCS's). This paper is a continuation of [10] and borrows the notation and the results from it. For other information see the standard references [6] and [7]. w Basic definitions and lemmas Recall that a pseudocomplement ed semilattice (PCS) is an algebra (S; A, *, 0) where (S; A, 0) is a A-semilattice with zero and * is the pseudocomplement ation and that the class of all PCS's is an equational class. Let S denote an arbitrary PCS and B(S) and N(S) denote respectively the set of closed (i,e. a**=a) elements and that of non-closed (i.e. a < a**) elements of S. It is well-known [5] that B(S) is both a Boolean algebra and a PCS-subalgebra of S. Con S denotes the congruence lattice of S with As and V~ (or simply A and V) as its least and greatest elements respectively; and the kernel of the homomorphism **:S ~ S,

Congruence permutability for algebras with pseudocomplementation

In this note we are concerned with the permutability of congruence relations on semilattices and lattices with pseudocomplementation. There are some results in the literature along these lines. For example, in (8) H. P. Sankappanavar characterises those pseudocomplemented semilattices whose congruence lattice is modular and employs the result in conjunction with the well-known fact that algebras with permuting congruences are congruence-modular to characterise those pseudocomplemented semilattices with permuting congruences. Our first result is a direct, short proof of his result. In (2), J. Berman shows that for all congruences on a distributive lattice L with pseudocomplementation to permute it is necessary and sufficient that D(L), the dense filter of L, be relatively complemented. Our second result is a generalisation of that result to an important equational class of lattices with pseudocomplementation which properly contains the modular lattices with pseudocomplementation.

Relatively pseudocomplemented semilattices amalgamate strongly

Relatively pseudocomplemented semilattices amalgamate strongly

Ideals and filters of pseudo-complemented semilattices

W. H. Cornish (2) has investigated congruences on pseudo-complemented distributive lattices and has identified those ideals (resp. filters) that are congruence kernels (resp. cokernels). In this paper we show that many of the principal results concerning congruence kernels and cokernels hold in a semilattice and therefore do not depend on distributivity, nor on the existence of unions.

Some remarks on certain classes of semilattices

In this paper the concept of a ∗‐semilattice is introduced as a generalization to distributive ∗‐lattice first introduced by Speed [1]. It is shown that almost all the results of Speed can be extended to a more eneral class of distributive ∗‐semilattices. In pseudocomplemented semilattices and distributive semilattices the set of annihilators of an element is an ideal in the sense of Grätzer [2]. But it is not so in general and thus we are led to the definition of a weakly distributive semilattice. In §2 we actually obtain the interesting corollary that a modular ∗‐semilattice is weakly distributive if and only if its dense filter is neutral. In §3 the concept of a sectionally pseudocomplemented semilattice is introduced in a natural way. It is proved that given a sectionally pseudocomplemented semilattice there is a smallest quotient of it which is a sectionally Boolean algebra. Further as a corollary to one of the theorems it is obtained that a sectionally pseudocomplemented semilattice with a dense element becomes a ∗‐semilattice. Finally a necessary and sufficient condition for a ∗‐semilattice to be a pseudocomplemented semilattice is obtained.

Congruence lattices of pseudocomplemented semilattices

Congruence lattices of pseudocomplemented semilattices