On axiom systems of pseudo-BCK algebras.
On BE-semigroups.
On Boolean modus ponens.
An abstract form of modus ponens in a Boolean algebra was suggested in [1]. In this paper we use the general theory of Boolean equations (see e.g. [2]) to obtain a further generalization. For a similar research on Boolean deduction theorems see [3].
On BPI Restricted to Boolean Algebras of Size Continuum
(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of...
On centers and state spaces of logics
On central atoms of Archimedean atomic lattice effect algebras
If element of a lattice effect algebra is central, then the interval is a lattice effect algebra with the new top element and with inherited partial binary operation . It is a known fact that if the set of central elements of is an atomic Boolean algebra and the supremum of all atoms of in equals to the top element of , then is isomorphic to a direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether is a bifull sublattice...
On closed classes of quasilinear functions
On collections of almost disjoint families
On complete -algebras
On complete-cocomplete subspaces of an inner product space
In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space is complete if and only if there exists a -additive state on , the orthomodular poset of complete-cocomplete subspaces of . We then consider the problem of whether every state on , the class of splitting subspaces of , can be extended to a Hilbertian state on ; we show that for the dense hyperplane (of a separable Hilbert space) constructed by P. Pták and...
On connections between information systems, rough sets and algebraic logic
In this note we remark upon some relationships between the ideas of an approximation space and rough sets due to Pawlak ([9] and [10]) and algebras related to the study of algebraic logic - namely, cylindric algebras, relation algebras, and Stone algebras. The paper consists of three separate observations. The first deals with the family of approximation spaces induced by the indiscernability relation for different sets of attributes of an information system. In [3] the family of closure operators...
On decidability of one class of boolean formulas
On effective presentations of formal concept lattices.
On Equational Classes of Algebraic Versions of Logic I.
On exact conditionals.
On extension properties for observables
On fields and ideals connected with notions of forcing
We investigate an algebraic notion of decidability which allows a uniform investigation of a large class of notions of forcing. Among other things, we show how to build σ-fields of sets connected with Laver and Miller notions of forcing and we show that these σ-fields are closed under the Suslin operation.
On free Boolean vectors.
On free -algebras
In the present paper we show that free -algebras can be constructed by applying free abelian lattice ordered groups.
On functions that cannot be mv-truth values in algebraic structures.
It is shown, in a general frame and playing with idempotency, that in order to have on a given lattice a Multiple Valued Logic preserving the lattice structure, the only t-norms and t-conorms allowing to modelize the truth values of a v b, a ^ b and a --> b are Min and Max, respectively, apart from ordinal sums.