Poset valued varieties.
On the collection of lattices determined by the same poset of meet-irreducibles.
Operators , and in the classes of -semigroups and -semirings.
Bisemilattice-valued fuzzy sets.
Weak congruence identities at 0.
Equivalent fuzzy sets
Necessary and sufficient conditions under which two fuzzy sets (in the most general, poset valued setting) with the same domain have equal families of cut sets are given. The corresponding equivalence relation on the related fuzzy power set is investigated. Relationship of poset valued fuzzy sets and fuzzy sets for which the co-domain is Dedekind-MacNeille completion of that posets is deduced.
Relative Complements in the Weak Congruence Lattice
Cardinality of height function’s range in case of maximally many rectangular islands - computed by cuts
We deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.
On p-semirings
A class of semirings, so called p-semirings, characterized by a natural number p is introduced and basic properties are investigated. It is proved that every p-semiring is a union of skew rings. It is proved that for some p-semirings with non-commutative operations, this union contains rings which are commutative and possess an identity.
A note on triangular schemes for weak congruences
Some geometrical methods, the so called Triangular Schemes and Principles, are introduced and investigated for weak congruences of algebras. They are analogues of the corresponding notions for congruences. Particular versions of Triangular Schemes are equivalent to weak congruence modularity and to weak congruence distributivity. For algebras in congruence permutable varieties, stronger properties—Triangular Principles—are equivalent to weak congruence modularity and distributivity.
A Note on Fuzzy Groups
Cut properties of resemblance
The resemblance relation is used to reflect some real life situations for which a fuzzy equivalence is not suitable. We study the properties of cuts for such relations. In the case of a resemblance on a real line we show that it determines a special family of crisp functions closely connected to its cut relations. Conversely, we present conditions which should be satisfied by a collection of real functions in in order that this collection determines a resemblance relation.
Congruences and homomorphisms on -algebras
The topic of the paper are -algebras, where is a complete lattice. In this research we deal with congruences and homomorphisms. An -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce -valued congruences, corresponding quotient -algebras and -homomorphisms and we investigate connections among these notions. We prove...
Poset-valued preference relations
In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of particular order reversing involution on the co-domain poset. We prove that for every set of alternatives...
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