a heyting algebra is a distributive lattice with implication and a dual $bck$-algebra is an algebraic system having as models logical systems equipped with implication. the aim of this paper is to investigate the relation of heyting algebras between dual $bck$-algebras. we define notions of $i$-invariant and $m$-invariant on dual $bck$-semilattices and prove that a heyting semilattice is equiva...

If P is an upper semilattice whose Hasse diagram is a tree and whose cartesian powers are Macaulay, it is shown that Hasse diagram of P is actually a spider in which all the legs have the same length.

The purpose of this paper is to characterize an ordered semigroup S in terms the properties associated B(S) all bi-ideals S. We show that a Clifford if and only semilattice. normal band both regular intra regular. For each subvariety V bands, we such ∈ V.

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive...

In this paper we initiate an investigation into the class of meet semilattices endowed with endomorphism. A consideration subdirectly irreducible algebras leads to a description subclass those (S;∧,k)(S;∧,k) in which (S;∧)(S;∧) is semilattice and kk endomorphism on SS characterised by property k⩾idSk⩾idS. We particularly show that such algebra if only it chain one following forms ⋯<aj&am...

We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D,≤T ,∨, ′〉 is decidable. We also prove that this result is not true about jusls with 0...

For a closure space (P,φ) with φ(∅) = ∅, the closures of open subsets of P , called the regular closed subsets, form an ortholattice Reg(P,φ), extending the poset Clop(P,φ) of all clopen subsets. If (P,φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, ...

We say that a 〈∨, 0〉-semilattice S is conditionally co-Brouwerian, if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e., x ≤ y for all 〈x, y〉 ∈ X × Y ), there exists z ∈ S such that X ≤ z ≤ Y , and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X , Y , and Z ...

In general, the tensor product, A ⊗ B, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If A ⊗ B is a capped tensor product, then A ⊗ B is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that A ⊗ B is a capped tensor product, for every lattice B with zero; we shall call such lattices ame...