Search

Let T X be the full transformation monoid over a finite set X , and fix some a ∈ T X of rank r. The variant T X a has underlying set T X , and operation f ⋆ g = f a g. We study the congruences of the subsemigroup P = Reg (T X a) consisting of all regular elements of T X a , and the lattice Cong (P) of all such congruences. Our main structure theorem ultimately decomposes Cong (P) as a specific subdirect product of Cong (T r) , and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice. [ABSTRACT FROM AUTHOR]

For a nonempty subset X of a ring R, the ring R is called X-semiprime if, given a ∈ R, aXa = 0 implies a = 0. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal L of a ring R, we offer a criterion for R to be L-semiprime. For a prime ring R, we characterizes Lie ideals L of R such that R is L-semiprime. Moreover, X-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied. [ABSTRACT FROM AUTHOR]

We generalize to d 0 -algebras a result of Riečanová about the decomposition of a D-lattice by means of a family of central elements. [ABSTRACT FROM AUTHOR]

In this note we present some examples of subdirect products of division rings. In particular, we answer two questions raised by Puczyłowski concerning which subdirect products of division rings are quasi-duo or weakly regular. [ABSTRACT FROM AUTHOR]

Let H ≤ S n be an intransitive group with orbits Ω 1 , Ω 2 , ... , Ω k. Then certainly H is a subdirect product of the direct product of its projections onto each orbit, H | Ω 1 × H | Ω 2 × ... × H | Ω k . Here we provide a polynomial time algorithm for computing the finest partition P of the H -orbits such that H = ∏ c ∈ P H | c and we demonstrate its usefulness in some applications. [ABSTRACT FROM AUTHOR]

We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated. [ABSTRACT FROM AUTHOR]

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$ , where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element. [ABSTRACT FROM AUTHOR]

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g : A → D , h : B → D , where A , B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C = { (a , b) ∈ A × B | g (a) = h (b) } is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation. [ABSTRACT FROM AUTHOR]

For n ≥ 2 and for a ring R, the notation Pn(R) means that an − a is nilpotent for all a ∈ R, and Qn(R) means that R/J(R) has identity xn = x and J(R) is nil, where J(R) is the Jacobson radical of R. In this paper, rings R for which Pn(R) holds are completely characterized for integers n up to 12, for n = 2k with k odd, and for n = 2kp, wherek > 0 and p = 3, 5, 7 or 9. For an arbitrary integer n ≥ 2, we prove that Pn(R) ⇔ Qn(R) for all rings R if and only if n is even with n ≡ 1 (mod 3), or n is odd with n ≢ 1 (mod 3) and n ≢ 1 (mod 8). [ABSTRACT FROM AUTHOR]

Some generalizations of BCK algebras (such as, from example, RM, BH, pre-BBBCC and pre-BCK algebras) are considered. It is proved that the set of all deductive systems of a pre-BBBCC algebra is an algebraic distributive lattice. Irreducible deductive systems and prime deductive systems of RM algebras are introduced and studied. In particular, the homomorphic properties of these deductive systems are provided. Finally, subdirect products of BH algebras by using prime deductive systems are investigated. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize a class of additively quasi regular semirings which are subdirect products of an idempotent semiring and a b-lattice of quasi skew-rings. [ABSTRACT FROM AUTHOR]

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometry f in a directed group G the relation f(UL(x, y) ∩ LU(x, y)) = UL(f(x), f(y)) ∩ LU(f(x), f(y)) is valid for each x, y ∈ G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is shown that each weak isometry in a 2-isolated directed group or in an abelian directed group is a composition of a subgroup symmetry and a right translation. It is also proved that stable weak isometries in a 2-isolated abelian directed group G are directly related to subdirect decompositions of the subgroup G2 = {2x; x ∈ G} of G. [ABSTRACT FROM AUTHOR]

The algebraic notion of a "congruence" seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph G = (V, E) being a pair (∼, ) of which ∼ is an equivalence relation on V and is a set of unordered pairs of vertices of G with a special relationship to ∼ and E. Kernels and quotient structures are used in this theory to develop homomorphism and isomorphism theorems which remind one of similar results in an algebraic context. We show that this theory can be applied to deliver structural decompositions of graphs into "factor" graphs having very special properties, such as the result that each graph, except one, is a subdirect product of graphs with universal vertices. In a final section, we discuss corresponding concepts and briefly describe a corresponding theory for graphs which have a loop at every vertex and which we call loopy graphs. They are in a sense more "algebraic" than simple graphs, with their meet-semilattices of all congruences becoming complete algebraic lattices. [ABSTRACT FROM AUTHOR]

A reduced Rickart ring is considered as a reduced ring R with an additional operation which associates to every element a∈R the single idempotent e such that the ideal eR is the right annihilator of a. We discuss some elementary properties of this operation, prove that a ring is reduced and Rickart if and only if it is isomorphic to an associate ring in the sense of I. Sussman (a certain subdirect product of domains with “enough” idempotents), and present several equational axiom systems for reduced Rickart rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce A-metric spaces (X, A, d), where X is a non empty set, A = (A, +, ≤, *) is a Representable Autometrized Algebra in [6], d : X × X → A is a function satisfying the formal properties of a distance function. We proved that any Ai-metric space (X, A, d) is congruent to a subdirect product of Aimetric spaces where each Ai is a totally ordered Representable autometrized algebra. [ABSTRACT FROM AUTHOR]