Domain Semirings United

Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.

Jauch–Piron states on quantum logics

We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.

Definable sets in boolean ordered o-minimal structures. II

Let (M, ≤,…) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

Hausdorff compactifications and zero-one measures

It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1, 2, 4, 12]). Using the technique developed in [5, 6], we find for any Tychonoff space X a Boolean algebra [Bscr ]X and a set [Lscr ]X of sublattices of [Bscr ]X having the following property: for any Hausdorff compactification cX of X there exists a (unique) LcX ∈ [Lscr ]X such that the maximal spectrum of LcX and the space of all u-regular zero-one measures on the Boolean subalgebra b(LcX) of [Bscr ]X, generated by LcX, are Hausdorff compactifications of X equivalent to cX. Let us give more details now.