Pseudocomplemented Lattices Research Articles

Priestley duality for demi-p-lattices

Priestley spaces of demi-p-lattices and almost-p-lattices are described in terms of Priestley spaces of pseudo-complemented distributive lattices. Characterizations of subvarieties of demi-p-lattices generated by a single finite subdirectly irreducible demi-p-lattice are given.

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.

Semi-DeMorgan algebras

Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of “partial diagrams” is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest variety of semi-DeMorgan algebras with the congruence extension property.

About the correspondences between dual heyting arrow lattices of a distributive lattice and its sublattices for relational biologic systems

Heyting and dual Heyting arrow operations relating non-comparable elements of finite relatively pseudo-complemented lattices being pseudo-Boolean algebras ( L) gave place to new structures named Heyting arrow ( L F ) and dual Heyting arrow lattices ( L F) (though sometimes they are only posets). They were used for analyzing qualitative relations in biological systems by means of isomorphisms relating the lattice elements with energy states identified through abstract relational concepts describing the system being represented. This paper considers the problem of connecting the poset L F with the posets L F (κ) corresponding to the epimorphic images L κ of a pseudo-Boolean lattice L.

Some results on L-fuzzy modules

Abstract An L -fuzzy quotient module of a module is redefined and a modular, complete pseudo-complemented lattice structure is given to the set of all L -fuzzy modules of a given module. Some isomorphisms between the lattices or posets of L -fuzzy modules are obtained. Finally a Galois connection is established between two lattices of L -fuzzy modules of homomorphic modules.

Finite pseudocomplemented lattices and ‘permutoedre’

We study finite pseudocomplemented lattices and especially those that are also complemented. With regard to the classical results on arbitrary or distributive pseudocomplemented lattices (Glivenko, Stone, Birkhoff, Frink, Grätzer, Balbes, Horn, Varlet,…), the finiteness property allows one to bring significant, more precise, details on the structural properties of such lattices. These results can especially be applied to the lattices defined by the ‘weak Bruhat order’ on a Coxeter group (and, for instance, to the lattice of permutations, called, in french, ‘le treillis permutoèdre’) and to the lattice of binary bracketings.

Bounded endomorphisms of free P-algebras

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

Les treillis pseudocomple´mente´s finis

Let L be a lattice with a least element denoted o; g(t) ∈ L is a meet pseudocomplement of t ∈ Lifx Λ t = o is equivalent to x ≤ g(t); L is meet pseudocomplemented if every element of L has a meet pseudocomplement. One defines dually the notion of join pseudocomplemented lattice. A lattice is pseudocomplemented if it is meet and join pseudocomplemented (such a lattice is sometimes called a double p -lattice, and the associated algebra a double p -algebra). These lattices have been intensively studied when they are distributive and infinite (Brouwer or Heyting lattices or algebras, Stone lattices or algebras). Examples of finite non-distributive pseudocomplemented lattices are the finite Coxeter groups endowed with the Bruhat weak ordering (especially, the ‘permutohedron’ lattice of Guilbaud and Rosenstiehl [22] is obtained with the symmetric group), and the lattice of binary bracketings (Friedman and Tamari [18]); these lattices are also complemented. We study the finite pseudocomplemented lattices, showing for instance that the pseudocomplements are obtained from the atoms and coatoms, that for complemented lattices meet pseudocomplementation is equivalent to join pseudocomplementation, and giving several results that clarify the structure of such lattices. These results can be applied especially to the permutohedron lattice and to the lattice of binary bracketings.

Semi-de Morgan algebras

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Lakser's characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]).In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

Semi-de Morgan algebras

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Lakser's characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]).In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

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.

Radicals in pseudocomplemented lattices

For a pseudocomplemented latticeL, we prove that the filter Dn(L), 1⩽n<ω, generated by then-strongly dense elements is contained in everyn-normal filter. Hence, Dn(L)=Gn(L)=Radn (L), where Gn(L) is the intersection of all n-normal filters, and Radn (L) is the intersection of alln-normal prime filters. Moreover, we prove that a prime filterP is n-normal iff Dn(L)=P. Consequently, for\(L \in \mathbb{B}_\omega \), we have Dn(L)=Gn(L)=Radn (L) and therefore\(L \in \mathbb{B}_n \) iff Radn(L)={1} (or iff Gn(L)={1}).

Projective distributive p-algebras

A characterization of finite (weak) projectives in an equational class of distributive pseudocomplemented lattices is given. In the class of all such lattices, a finite lattice is projective if and only if its poset of join–irreducible elements forms a semilattice in which the minimal elements below the join of x and y are exactly the mininal elements below x or y. A similar condition works for any equational subclass.

Generalized relatively pseudocomplemented lattices

Generalized relatively pseudocomplemented lattices

Nontrivially pseudocomplemented lattices are complemented

Nontrivially pseudocomplemented lattices are complemented.

The triple method and free distributive pseudocomplemented lattices

The triple method and free distributive pseudocomplemented lattices

Implicational Classes of Pseudocomplemented Distributive Lattices

Implicational Classes of Pseudocomplemented Distributive Lattices

A Sheaf Representation of Distributive Pseudocomplemented Lattices

A Sheaf Representation of Distributive Pseudocomplemented Lattices

Principal congruences of $p$-algebras and double $p$-algebras

Principal congruence of pseudocomplemented lattices (= palgebras) and of double pseudocomplemented lattices (= double p-algebras), i.e. pseudocomplemented and dual pseudocomplemented ones, are characterized.

A sheaf representation of distributive pseudocomplemented lattices

The main result of this paper shows that a distributive pseudocomplemented lattice ( L ; ∨ , ∧ , ∗ , 0 , 1 ) (L; \vee , \wedge {,^ \ast },0,1) , considered as an algebra of type ⟨ 2 , 2 , 1 , 0 , 0 ⟩ \langle 2,2,1,0,0\rangle , can be represented as the algebra of all global sections in a certain sheaf. The stalks are the quotient algebras L / Θ ( O ( P ) ) L/\Theta (O(P)) , where P P is a prime ideal in L L . The base space is the set of prime ideals of L L equipped with the topology whose basic open sets are of the form P : P P:P prime in L , x ∗ ∗ ∉ P L,{x^{ \ast \ast }} \notin P for some x ∈ L x \in L .