Distributive and completely distributive lattice extensions of ordered sets

It is known that a poset can be embedded into a distributive lattice if, and only if, it satisfies the prime filter separation property. We describe here a class of “prime filter completions” for posets with the prime filter separation property that are completely distributive lattices generated by the poset and preserve existing finite meets and joins. The free completely distributive lattice generated by a poset can be obtained through such a prime filter completion. We also show that every completely distributive completion of a poset with the prime filter separation property is representable as a canonical extension of the poset with respect to some set of filters and ideals. The connections between the prime filter completions and canonical extensions are described and yield the following corollary: the canonical extension of any distributive lattice is the free completely distributive lattice generated by the lattice. A construction that is a variant of the prime filter completion is given that can be used to obtain the free distributive lattice generated by a poset. In addition, it is shown that every distributive lattice extension of the poset can be represented by such a construction. Finally, we show that a poset with the prime filter separation property and the free distributive lattice generated by it generates the same free completely distributive lattice.

A study of stabilizers in triangle algebras

In this paper, we introduce the notions of stabilizer of a subset and the stabilizer of a subset with respect to another one in triangle algebras and study them in details. It is shown that the stabilizer of a subset and stabilizer of an interval valued residuated lattice filter (IVRL-filter) with respect to another IVRL-filter are IVRL-filters. We state and prove some theorems which determine some properties of this stabilizers in triangle algebras. Also, we prove that in linearly ordered triangle algebras, stabilizer of a set is an IVRL-extended prime filter. Finally, we consider the influence of stabilizers on product and quotient triangle algebras.

Dual annihilator filters of commutative BE-algebras

Some properties of dual annihilator filters of commutative [Formula: see text]-algebras are studied. It is proved that the class of all dual annihilator filters of a BE-algebra is a complete Boolean algebra. A set of equivalent conditions is derived for every prime filter of a commutative [Formula: see text]-algebra to become a maximal filter.

Research of Unary Lattice Implication Algebra Equations on Interval Sets

The concepts are re-defined on the interval sets which are filter, prime filter, LI-ideal, dual atom and convex sub-lattice in the lattice implication algebra. Three basic unary lattice implication algebra equations on the interval sets are researched. The necessary and sufficient conditions for existence of solutions for the equations are presented. And some properties of equation sets also are given.

The prime filter theorem of lattice implication algebras

Using a special setx−1F, we give an equivalent condition for a filter to be prime, and applying this result, we provide the prime filter theorem in lattice implication algebras

The effects of pulsatile flow on the leukocyte depleti ng qualities of the Pall LG6 leukocyte depleting arterial line filter: a laboratory investigation

The Pall LG6 arterial line filter has, in a previous publication, demonstrated its inherent leukocyte depleting qualities. This initial study was however carried out under continuous flow conditions. The present study was designed to assess the effectiveness of the LG6 filter in performing this leukocyte removal function under the more dynamic conditions of pulsatile flow. In addition to leukocyte depletion, the general blood handling and degree of energy absorbtion associated with the LG6 and Stat-Prime filters was also assessed. The results demonstrated that the LG6 filter was unaffected by the flow regime employed in terms of leukocyte removal and platelet depletion. There was a higher level of measured haemolysis associated with the use of pulsatile rather than nonpulsatile flow, however, this was the case with both filter types and was not found to be the case when generated values were computed. The LG6 filter absorbed more energy than the Stat-Prime filter as reflected by energy equivalent pressure (EEP) measurement, but this difference did not reach a level which was considered to be clinically significant.

The ideal lattice of an MS-algebra

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

0-Distributive and P-Uniform Semilattices

A counter-example is provided to the conjecture of Y. S. Pawar and N. K. Thakare that a semilattice S with 0 is 0-distributive if and only if for each filter F and each ideal I such that F ∩ I = Ø, there exists a prime filter containing F and disjoint from I. This shows that 0-distributivity is not equivalent to weak distributivity. A characterization is also given of finite P-uniform semilattices in terms of 0-distributivity.