We deene conndence relations on Boolean lattices, which can be interpreted as ordinal representations of uncertainty or information. The set of the conndence 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 an ...

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...

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...

This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids (RSs). An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has this closure property if its finitely sub...

We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...

We prove that for every distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) ≤ a ∨ b, then there are a pos...

We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...

The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra H considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice DedH of all deductive systems on H and every maximal deductive system is prime. Complements and relative complements of DedH are characterized as the so called annihila...

We investigate the class of reflexive graphs that admit semilattice polymorphisms, and in doing so, give an algebraic characterisation of chordal graphs. In particular, we show that a graph G is chordal if and only if it has a semilattice polymorphism such that G is a subgraph of the comparability graph of the semilattice. Further, we find a new characterisation of the leafage of a chordal grap...