Centrally orthocomplete effect algebras

Abstract

Motivated by the theory of Loomis dimension lattices, we generalize the notion of a hull mapping to an arbitrary effect algebra (EA). Using hull mappings, we identify certain special types of elements in an EA, including generalizations of the invariant elements and of the simple elements in a dimension lattice. We introduce and study a new class of effect algebras, called centrally orthocomplete effect algebras (COEAs), satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. We show that COEAs admit a central cover mapping and we develop the basic theory of direct decomposition of COEAs.

Hull mappings and dimension effect algebras

“…The center of E, which we shall denote by Γ(E), is defined to be the set of all central elements in E. It turns out that Γ(E) is a sub-EA of E and, as such, it is boolean; moreover, if c, d ∈ Γ(E), then the supremum and infimum of c and d in the boolean algebra Γ(E) are also the supremum c ∨ d and the infimum c ∧ d of c and d in E [3, Theorem 1.9.14]. Also by [3,Lemma 1.9.12], [6,Theorem 4.4 (ii)], and [6,Corollary 4.8], if c ∈ Γ(E), and e ∈ E, then (i) c ∧ e and c ⊥ ∧ e exist in E,…”

“…If p ∈ E, then by [6,Lemma 4.9], the mapping c → p ∧ c for c ∈ Γ(E) is a boolean homomorphism of the boolean EA Γ(E) into the center Γ(…”

“…First, to extend the type decomposition theory for effect algebras (EAs) developed in [6] and [7] to the case of a In Section 7, we introduce our definition of a dimension effect algebra (DEA), i.e., an orthocomplete EA E equipped with a dimension equivalence relation (DEA) ∼. We show that a DEA E determines a unique hull mapping η on E such that the invariant elements with respect to ∼ coincide with the η-invariant elements (Theorem 7.11).…”

Quotients of dimension effect algebras

“…As D is an orthogonal set in Γ(E), [13,Corollary 6.7] implies that e ∧ h = e∧ D = {e∧d : d ∈ D}. Clearly, e∧h is an upper bound in E for (e∧h i ) i∈I .…”

Type Decomposition of a Pseudoeffect Algebra

“…Let S denote the set of all subcentral elements of E, B the set of all boolean elements of E, and H the set of all monads in E. As in [11], it can be shown that S is a type-determining set with [A] ⊆ S, B is a strongly type-determining set with (i) K is closed under passage to direct summands, that is, if H ∈ K and h ∈ (H ),…”