On the notion of ranked complementedness in a lattice

Not every element in a lattice has a complement. In this paper we introduce a notion of ranked complement, which depends on a natural number [Formula: see text], so that for every element [Formula: see text] in a lattice with finite height there exists [Formula: see text] such that [Formula: see text] has a complement of rank [Formula: see text]. One of the main results we establish is that in a modular lattice having finite height, every element has a complement of rank less than [Formula: see text] if and only if there is a chain [Formula: see text] of elements such that each interval [Formula: see text] is a complemented lattice.

The lattices of families of regular sets in topological spaces

The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

Stabilizers in EQ-algebras

The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where Str(F, G) is the relative pseudo-complemented of F with respect to G. These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.

Single identities forcing lattices to be Boolean

In this note we characterize Boolean algebras among lattices of type (2, 2, 1) with join, meet and an additional unary operation by means of single two-variable respectively three-variable identities. In particular, any uniquely complemented lattice satisfying any one of these equational constraints is distributive and hence a Boolean algebra.

On annihilators in BL-algebras

In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,{0}, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then $J_I^ \bot $ is the relative pseudo-complement of J with respect to I in the ideal lattice (I(L), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α-ideal and give a notation E(I ). We show that (E(I(L)), ∧E, ∨E, E(0), E(L) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.

β-FILTERS OF MS-ALGEBRAS

The concepts of boosters and β-filters are introduced in an MS-algebra and the β-filters are characterized in terms of boosters. It is then proved that the lattice of β-filters is isomorphic to the ideal lattice of the lattice of boosters. A set of equivalent conditions are derived for the lattice of boosters to become a relatively complemented lattice.

Mean ergodic operators and reflexive Fréchet lattices

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

Congruence Class Sizes in Finite Sectionally Complemented Lattices

The congruences of a finite sectionally complemented lattice L are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence Θ of L is from being uniform, we introduce Spec Θ, the spectrum of Θ, the family of cardinalities of the congruence classes of Θ. A typical result of this paper characterizes the spectrum S = (mj | j < n) of a nontrivial congruence Θ with the following two properties:

The Tensor Product Formula for Reflexive Subspace Lattices

We give a characterisation of where and are subspace lattices with commutative and either completely distributive or complemented. We use it to show that Lat is a CSL algebra with a completely distributive or complemented lattice and is any operator algebra.

The Macneille Completion of a Uniquely Complemented Lattice

Problem 36 of the third edition of Birkhoff's Lattice theory [2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented.