A Heyting effect algebra (HEA) is a lattice-ordered effect algebra that is at the same time a Heyting algebra and for which the Heyting center coincides with the effect-algebra center. Every HEA is both an MV-algebra and a Stone-Heyting algebra and is realized as the unit interval in its own universal group. We show that a necessary and sufficient condition that an effect algebra is an HEA is that its universal group has the central comparability and central Rickart properties.

We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.

In this paper, by considering the notions of effect algebra and product effect algebra, we define the concept of effect module. Then we investigate some properties of effect modules, and we present some examples on them. Finally, we introduce some topologies on effect modules.

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.

A Heyting effect algebra is a lattice-ordered effect algebra that is at the same time a Heyting algebra and for which the Heyting center coincides with the effect-algebra center. Every HEA is both an MV-algebra and a Stone-Heyting algebra and is realized as the unit interval in its own universal group. We show that a necessary and sufficient condition that an effect algebra is an HEA is that its universal group has the central comparability and central Rickart properties.

We review the fact that an MV-algebra is the same thing as a lattice-ordered effect algebra in which disjoint elements are orthogonal. An HMV-algebra is an MV-effect algebra that is also a Heyting algebra and in which the Heyting center and the effect-algebra center coincide. We show that every effect algebra with the generalized comparability property is an HMV-algebra. We prove that, for an MV-effect algebra E, the following conditions are mutually equivalent: (i) E is HMV, (ii) E has a (...) center valued pseudocomplementation, (iii) E admits a central cover mapping γ such that, for all p, q∈E, p∧q=0⇒γ(p)∧q=0. (shrink)

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are (...) MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr. (shrink)