A functional representation of the hyperspace monad, based on the semilattice structure of function space, is constructed.

Given a Banach space X, denote by SPw(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SPw(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphi...

We begin by recalling the general theory of adjoints on finite semilattices. A finite join semilattice with 0 is a lattice, with the naturally induced meet operation. Thus a finite lattice S can be regarded as a semilattice in two ways, either S = 〈S,+, 0〉 or S = 〈S,∧, 1〉. Given a (+, 0)-homomorphism g : S → T , define the adjoint h : T → S by h(t) = ∑ {s ∈ S : gs ≤ t} so that gs ≤ t iff s ≤ ht...

We describe the semilattice of ordered compactifications ofX×Y smaller than βoX×βoY whereX and Y are certain totally ordered topological spaces, and where βoZ denotes the Stone–Čech orderedor Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications ofX×Y smaller than βoX×βoY for arbitrary totally ordered topol...

Let A and B be lattices with zero. The classical tensor product, A ⊗ B, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: A ⊗ B is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of c...

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

We study all possible constant expansions of the structure dense meet-tree ⟨М; <, П⟩ [3]. Here, a is lower semilattice without least and greatest elements. An example this with expansion theory that has exactly three pairwise non-isomorphic countable models [6], which good in context Ehrenfeucht theories. by using general classification complete theories [7], as well description specificity ...