Narrow Results

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.

In this paper we describe the lattice of preradicals over any local uniserial ring (equivalently, any local artinian principal ideal ring). This lattice is isomorphic to a lattice of binary sequences of length n, where n is the length of the composition series for the ring, as a left and as a right module. We prove some of its properties: it is finite having cardinality 2ⁿ, it is distributive, self-dual and it is graded having rank . We also describe the correspondent posets of irreducible, join-irreducible, prime and coprime elements.

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if $B$ is an $S$-semialgebra, then under some conditions, the set of zero-divisors $Z(B)$ of $B$ is the union of the extended maximal primes of $Z(S)$. Finally, we prove a generalization of Eisenstein’s irreducibility criterion.

Attribute reduction of formal context is a crucial reseach issue in formal concept analysis. In this paper, based on the meet-irreducible elements and join-irreducible elements of concept lattice, two kinds of attribute reductions of formal context are proposed, which are called MI-attribute reduction and JI-attribute reduction. Subsequently, we discuss the relationships among them and two existing attribute reductions of formal context, lattice-based attribute reduction and granular reduction. Consequently, we find that the MI-attribute reduction and lattice-based attribute reduction are identical. For JI-attribute reduction, the judgement theorems of JI-consistent attribute sets are obtained. Finally, by using the discernibility attribute sets, a method of computing all JI-attribute reducts of a formal context is presented.