For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

My primary and current research work is in the general spirit of algebras of logic, lattice theory, hyper structures and applications. A partially ordered set, or poset for short is a pair (P, ≤) where P is a set and ≤ a partial order on P. A poset (P, ≤) is called a lattice if every pair x, y ∈ P has a least upper bound x ∨ y and a greatest lowest bound x ∧ y in P. A lattice is bounded if it h...

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ be a homomorphism of [a, b] onto [c, d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extens...

The notions of a S fuzzy ∧ sub semi lattice, a S fuzzy ideal and a S fuzzy prime ideal of a bounded lattice with truth values in a bounded ∧ sub semi lattice S are introduced which generalize the existing notions with truth values in a unit interval of real numbers. Finally, S fuzzy prime ideal theorem is proved. 2010 AMS Classification: 03G10, 46H10, 06D50, 08A72

We prove that when a pre-independence space satisfies some natural properties, then its cyclic flats form a bounded lattice under set inclusion. Additionally, we show that a bounded lattice is isomorphic to the lattice of cyclic flats of a pre-independence space. We also prove that the notion of cyclic width gives rise to dual-closed and minorclosed classes of B-matroids. Finally, we find a dif...

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topolo...

Abstract. Let L be a lattice and M a bounded distributive lattice. Let ConL denote the congruence lattice of L, P (M) the Priestley dual space of M , and L (M) the lattice of continuous order-preserving maps from P (M) to L with the discrete topology. It is shown that Con(L ) ∼= (ConL) P (ConM) Λ , the lattice of continuous order-preserving maps from P (ConM) to ConL with the Lawson topology. V...

Let L be a bounded distributive lattice. For k 1, let Sk (L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean functions. The lattices that can arise as Sk (L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lat...