We define confidence relations on Boolean lattices, which can be interpreted as ordinal representations of uncertainty or information. The set of the confidence relations on a given Boolean lattice can be ordered by set inclusion and thus is shown to form a complete meet-semilattice. We investigate and identify the maximal elements of this structure. Moreover, we prove that it is in particular ...

Weak relative pseudocomplementation on a meet semilattice S is a partial operation ∗ which associates with every pair (x, y) of elements, where x ≥ y, an element z (the weak pseudocomplement of x relative to y) which is the greatest among elements u such that y = u ∧ x. The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplemen...

We study how the existence in an algebraic lattice L of a chain of a given type is reflected in the join-semilattice K(L) of its compact elements. We show that for every chain α of size κ, there is a set B of at most 2 join-semilattices, each one having a least element such that an algebraic lattice L contains no chain of order type I(α) if and only if the join-semilattice K(L) of its compact e...

Weak relative pseudocomplementation on a meet semilattice S is a partial operation ∗ which associates with every pair (x, y) of elements, where x ≥ y, an element z (the weak pseudocomplement of x relative to y) which is the greatest among elements u such that y = u ∧ x. The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplemen...

We show that for every quasivariety K of relational structures there is a semilattice S with operators such that the lattice of quasiequational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+, 0, F). It is known that if S is a join semilattice with 0, then there is a quasivariety Q such that the lattice of theories of Q is isomorphic to Con(S,+, 0) (w...

In this paper, the concept of semiprimary fuzzy ideals of an ordered semigroup S is introduced. Some characterizations for an ordered semigroup S to be a semilattice of archimedean (rarchimedean, t-archimedean) ordered subsemigroups are given by some binary relations on S and the fuzzy radical of fuzzy ideals (fuzzy right ideals, fuzzy bi-ideals) of S. Furthermore, some characterizations for an...

We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join-semilattice of compact elements of L, is well-founded and contains neither [ω], nor Ω(ω∗) as a join-subsemilattice. As an immediate corollary, we get that...

For a class V of algebras, denote by Conc V the class of all (∨, 0)semilattices isomorphic to the semilattice Conc A of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1;V2) the smallest cardinality of a (∨, 0)-semilattice in Conc V1 which is not in Conc V2 if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that i...

We study completely π-regular semigroups admitting a decomposition into a semilattice of σn-simple semigroups, and describe them in terms of properties of their idempotents. In the general case, semigroups admitting a decomposition into a semilattice of σn-simple semigroups were characterized by M. Ćirić and S. Bogdanović in [3] (see Theorem 1 below), in terms of paths of length n in the graph ...