In this paper we shall give some results on irreducible deductive systems in BCK-algebras and we shall prove that the set of all deductive systems of a BCK-algebra is a Heyting algebra. As a consequence of this result we shall show that the annihilator of a deductive system is the the pseudocomplement of . These results are more general than that the similar results given by M. Kondo in [7].
Using the polytopes defined in an earlier paper, we show in this paper the existence of degeneration of a large class of Schubert varieties of to toric varieties by extending the method used by Gonciulea and Lakshmibai for a miniscule to Schubert varieties in .
Based on a completely distributive lattice , degrees of compatible -subsets and compatible mappings are introduced in an -approximation space and their characterizations are given by four kinds of cut sets of -subsets and -equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible -subsets and compatible degrees of -subsets. Finally, the notion of complete -sublattices is introduced and it is shown that the...
In an algebraic frame the dimension, , is defined, as in classical ideal theory, to be the maximum of the lengths of chains of primes , if such a maximum exists, and otherwise. A notion of “dominance” is then defined among the compact elements of , which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of , including the frames and of -elements and -elements, respectively. The more concrete illustrations...
This paper continues the investigation into Krull-style dimensions in algebraic frames. Let be an algebraic frame. is the supremum of the lengths of sequences of (proper) prime elements of . Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of in terms of the dimensions of certain boundary quotients of . This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...
The notion of bounded commutative residuated -monoid (-monoid, in short) generalizes both the notions of -algebra and of -algebra. Let be a -monoid; we denote by the underlying lattice of . In the present paper we show that each direct...
Let be an infinite cardinal. Let be the class of all lattices which are conditionally -complete and infinitely distributive. We denote by the class of all lattices such that is infinitely distributive, -complete and has the least element. In this paper we deal with direct factors of lattices belonging to . As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class .
In this paper we deal with the relations between the direct product decompositions of a pseudo -algebra and the direct product decomposicitons of its underlying lattice.