Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axi...

1. Introduction IT is well known that there are a number of sets of axioms for Boolean algebra. In this paper, I give a set of five independent axioms using conditioned disjunction as the ternary operation which is postulated. The ternary operation is most naturally defined on the set 2 n —this is done in §2. The theorem is stated in §3 together with a slight generalization of this result; the ...

As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary...

The concept of the direct product finite family \(\textit{B}\)-algebras is introduced by Lingcong and Endam [J. A. V. Lingcong, J. C. Endam, Int. Algebra, \(\textbf{10}\) (2016), 33--40]. In this paper, we introduce infinite BCC-algebras prove that it a dual (dBCC-algebras), call external dBCC-algebra induced BCC-algebras, which general in sense Endam. We find result special subsets BCC-algebra...

Let G be an Abelian group and let μ : A → G and ν : B → G be finitely additive measures (charges) defined on fields A and B of subsets of a set X. It is assumed that μ and ν agree on A∩B, i.e. they are consistent. The existence of common extensions of μ and ν is investigated, and conditions on A and B facilitating such extensions are given. 0. Introduction. We consider the following problem: Le...

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra Aλ whose every infinite homomorphic image is of cardin...

We define Boolean algebras over nominal sets with a functionsymbol Nmirroring the N‘fresh name’ quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality.

We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in Martin-L type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how ...

For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grzn (n ≥ 1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer ...

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-alg...