Compact generation in partially ordered sets

Abstract

Several" "classical" results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own "up-completion" i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.

Sharp partial closure operator

“…Next we deal with the set counterpart of partial closure operators. A partial closure system (in the literature known also as a centralized system, e. g. [5,6]) is a family F of subsets of a nonempty set S satisfying:…”

“…Completions of a poset are also a subject of [8]. Extensive research of generalizations of closure systems and related posets, together with the corresponding properties is done by M. Erné (e.g., [5,6]). In [7], the lattice of all Dedekind-MacNeille completions of posets with the fixed join-irreducibles is investigated.…”

On lattices and their ideal lattices, and posets and their ideal posets

“…First, some terminology and notation. Generalizing slightly the language of [7], let us call an element x of a poset P compact if for every directed subset S ⊆ P which has a least upper bound S in P, and such that S ≥ x, there is some y ∈ S which already majorizes x. For P any poset, the compact elements of id(P ) are the principal ideals.…”

“…It is interesting to compare the situation of the preceding theorem with what we get if we start with any poset P with a nonprincipal ideal I, and consider the posets (7) P → id(P ) → id 2 (P ) → . .…”

“…On the other hand, the ideal of id(P ) generated by that set, since it consists of elements < b 1 , can be regarded as an element b 2 ∈ id 2 (P ) which is < d id(P ) (b 1 ); this element in turn will strictly majorize all elements of d id(P ) d P (I), and so the ideal generated by that set in id 2 (P ) will be an element of id 3 (P ) which is < d id 2 (P ) (b 2 ); and so on. Letting P ∞ denote the direct limit of (7), and writing d ∞,n : id n (P ) → P ∞ for the induced maps to that object, we get a descending chain d ∞,1 (b 1 ) > d ∞,2 (b 2 ) > . .…”