Gelfand, Retakh, Serconek and Wilson, in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra ...

In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Gröbner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations....

Neutrosophic set is a new mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. Pseudo-BCI algebra is a kind of non-classical logic algebra in close connection with various non-commutative fuzzy logics. Recently, we applied neutrosophic set theory to pseudo-BCI algebras. In this paper, we study neutrosophic filters in pseudo-BCI algebras. The concepts...

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

Many interesting geometric structures can be defined by specifying a smooth subbundle V of the complexified tangent bundle of the underlying smooth manifold, subject to an integrability condition. Examples include foliations, complex structures, and CR structures. In general such structures are called involutive, or formally integrable, and their study has been the starting point of far-reachin...

Let A be an associative algebra over C and L an invariant linear functional on it (trace). Let ω be an involutive antiautomorphism of A such that L(ω(a)) = L(a) for any a ∈ A. Then A admits a symmetric invariant bilinear form 〈a, b〉 = L(aω(b)). For A = U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis whose elements turned out to be, essentiall...

At first, we recall the basic concept, By a residual lattice is meant an algebra ) 1 , 0 , , , , , ( o ∗ ∧ ∨ = L L such that (i) ) 1 , 0 , , , ( ∧ ∨ = L L is a bounded lattice, (ii) ) 1 , , ( ∗ = L L is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: y z y x = ∗ ∨ ) ( if and only if y x z y o ≤ ≤ Let us note [7] that y x ∨ is the greatest element of the set y z y x ...

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a re...

For an algebra A of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that A separates points in the sense that for each distinct pair x, y ∈ X, there exists an f ∈ A such that f(x) 6= f(y). If A does not separate points, it is known that there exists an algebra  on a compact Hausdorff space (X̂, τ̂) that does separate points such that th...

We compute the involutive Heegaard Floer homology of the family of threemanifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orien...