Relative polars in ordered sets
In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of -polars are studied. Connections between -polars and prime ideals, especially in distributive sets, are found.
Relative Pseudo-Complements, Join-Extensions, and Meet-Retractions.
Relatively additive states on quantum logics
In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous...
Relatively complemented ordered sets
We investigate conditions for the existence of relative complements in ordered sets. For relatively complemented ordered sets with 0 we show that each element b ≠ 0 is the least one of the set of all upper bounds of all atoms contained in b.
Relatively free lattices
Relatively pseudocomplemented directoids
The concept of relative pseudocomplement is introduced in a commutative directoid. It is shown that the operation of relative pseudocomplementation can be characterized by identities and hence the class of these algebras forms a variety. This variety is congruence weakly regular and congruence distributive. A description of congruences via their kernels is presented and the kernels are characterized as the so-called -ideals.
Relatively pseudocomplemented Hilbert algebras
We characterise those Hilbert algebras that are relatively pseudocomplemented posets.
Relatively pseudocomplemented posets
We extend the notion of a relatively pseudocomplemented meet-semilattice to arbitrary posets. We show some properties of the binary operation of relative pseudocomplementation and provide some corresponding characterizations. We show that relatively pseudocomplemented posets satisfying a certain simple identity in two variables are join-semilattices. Finally, we show that every relatively pseudocomplemented poset is distributive and that the converse holds for posets satisfying the ascending chain...
Remark on partially ordered sets, universal algebras and semigroups
Remarks about measures on orthomodular posets
Remarks on affine complete distributive lattices
We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that ℚ ∩ [0, 1] is initial in the class of affine complete lattices.
Remarks on cellularity in products
Remarks on chain conditions in products
Remarks on equational theories of semilattices with operators
Remarks on Ideals in Join-Semilattices.
Remarks on ideals in lower-bounded dually residuated lattice-ordered monoids
Lattice-ordered groups, as well as -algebras (pseudo -algebras), are both particular cases of dually residuated lattice-ordered monoids (-monoids for short). In the paper we study ideals of lower-bounded -monoids including -algebras. Especially, we deal with the connections between ideals of a -monoid and ideals of the lattice reduct of .
Remarks on isomorphisms of regressive transformation semigroups.
Remarks on measurable Boolean algebras and sequential cardinals
The paper offers a generalization of Kalton-Roberts' theorem on uniformly exhaustive Maharam's submeasures to the case of arbitrary sequentially continuous functionals. Applying the result one can reduce the problem of measurability of sequential cardinals to the question whether sequentially continuous functionals are uniformly exhaustive.
Remarks on powers of lattices
Remarks on products of σ-ideals