Related papers: Multiplicative lattices with absorbing factorizati…
We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several…
We study separation axioms for $X$-top-lattices (i.e. lattices $L$ for which a given subset $X\subseteq L\backslash \{1\}$ admits a \emph{Zariski-like topology}). Such spaces are $T_{0}$ and usually far away from being $T_{2}.$% We give…
Motivated by situations in which the removal of a zero (a.k.a., an absorbing element) from a semigroup yields a subsemigroup with another zero, sets of quasi-zeros (a.k.a., quasi-absorbing elements) are introduced as well as primitive…
Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…
We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…
An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary…
If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a…
We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has…
As the main achievement of the paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, the paper contains the following three observations. First, every coatomless three-generated…
Each finite algebra $\mathbf A$ induces a lattice~$\mathbf L_{\mathbf A}$ via the quasi-order~$\to$ on the finite members of the variety generated by~$\mathbf A$, where $\mathbf B \to \mathbf C$ if there exists a homomorphism from $\mathbf…
A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^*a; a\text{ is invertible with }a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive and…
We say that a countable discrete group action $\alpha$ on a C$^*$-algebra $A$ is \textit{$\mathcal{W}$-absorbing} if there exist a C$^*$-algebra $B$ and an action $\beta$ on $B$ such that $\alpha$ is cocycle conjugate to $\beta\otimes…
In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$ associated to a well-generated complex reflection group $W$. Chief component of this was the trivialization theorem, a fundamental correspondence…
In the first section of the present work, we introduce the concept of pseudocomplementation for semirings and show semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some…
We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible…
It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic…
In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in…
This is Part A of four Parts dedicated to modular lattices of finite length. It builds on 1992 notes of the author (available on ResearchGate), and in so doing heeds a wish of the late Gian-Carlo Rota. Part A is in fairly final form and…
We study several separation axioms for $X$-top-lattices (i.e. a lattice $L$ for which a given subset $X\subseteq L\backslash \{1\}$ admits a \emph{% Zariski-like topology}). Such spaces are $T_{0}$ and usually far away from being $T_{2}.$…
We survey results concerning special elements of nine types (modular, lower-modular, upper-modular, cancellable, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and certain…