Weakly distributive semilattices

A note on annihilators in distributive nearlattices

“…Let A be a nearlattice. Then A is distributive if and only if satisfies either of the following identities: For distributive nearlattices we have the following lemma which characterizes the generated filters and can be deduced from the results given in [9]. Lemma 1.…”

“…A nearlattice is a join-semilattice in which every principal filter is a lattice. The class of nearlattices forms a variety that has been studied in [9] and [11] by Cornish and Hickman, and in [4], [6] and [7] by Chajda, Kolařík, Halaš and Kühr. In [2] the authors showed that the axiom systems given in [11] and [4] are dependent and that the variety of nearlattices is 2-based.…”

Mildly distributive semilattices

“…It is possible to prove this by direct computation. However, by Theorem 3.8, a $ f (S) = f"(S), the lattice of all finitely generated ideals of S, and each strong join partial homomorphism from S is join partial, so a computational proof would mimic the details of Theorem 1.3 from Cornish and Hickman [4], and so we refer the reader to that paper. A semilattice congruence on S, which satisfies (i) will be called join partial.…”

Closure Operators, Frames and Neatest Representations

“…Schein's 3-distributivity can be generalised to the concept of α-distributivity for cardinals α (see Definition 3.1). This notion has been studied when α = n < ω [11], when α = ω [4,20] and when α is any regular cardinal [12]. Note that if m, n ≤ ω with m < n, then n-distributivity trivially implies m-distributivity, but the converse is not true [13].…”