Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Hierarchies of tree series transformations revisited

Abstract: Tree series transformations computed by polynomial top-down and bottom-up tree series transducers are considered. The hierarchy of tree series transformations obtained in [Fulop, Gazdag, Vogler: Hierarchies of Tree Series Transformations. Theoret. Comput. Sci. 314(3), p. 387–429, 2004] for commutative izz-semirings (izz abbreviates idempotent, zero-sum and zero-divisor free) is generalized to arbitrary positive (i.e., zero-sum and zero-divisor free) commutative semirings. The latter class of semirings includes prominent examples such as the natural numbers semiring and the least common multiple semiring, which are not members of the former class.

Monoids, Boolean Algebras, Materially Ordered Sets

Abstract: In this paper, the interplay between certain mathematical structures is elucidated. First, it is shown that there is a one-to-one correspondence between bounded half-lattices and commutative idempotent monoids (c.i.-monoids). Adding certain additional structural ingredients and axioms, such c.i.-momoids become Boolean algebras. There is a non-trivial one-to-one correspondence between these and what we call materially ordered sets, which are half -lattices that satisfy certain additional axioms. Such materially ordered sets can serve as mathematical models for certain physical systems. The correspondence between materially ordered sets and Boolean algebras can be used to show, for example, that the law of action and reaction (Newton’s third law) is not an independent axiom but a consequence of fundamental balance laws. 0. Mathematical Structures A mathematical structure is described by prescribing ingredients and postulating axioms, which are conditions that the ingredients are assumed to satisfy. In most cases, one starts with a single set and endows it with structure by specifying ingredients that are entities involving constructions from this given set. An isomorphism between two structures of the same type is an invertible mapping between the underlying sets that induces a correspondence between the ingredients. An automorphism is an isomorphism from the structured set to itself. Given a set S endowed with a specified structure and an arbitrary invertible mapping from S to a set T , one can use this mapping to transport the structure from S to T by transporting the ingredients of S to T . The axioms for T are then automatically satisfied. In some of the cases, the set T may coincide with S and then S acquires a second structure of the same type. The mapping is an automorphism only if this second structure coincides with the given one. These considerations will be illustrated by the content of the remainder of this paper. Given a set S we define Sub S to be the set of all subsets of S. Let f : A → B be a mapping with domain A and codomain B. The image mapping of f is the mapping f> : Sub A → Sub B defined by f>(U) := {f(x) | x ∈ U} for all U ∈ Sub A. (1) Let S ∈ Sub A and T ∈ Sub B be such that f>(S) ⊆ T . Then the adjustment f |S : S → T of f is defined by f |S (x) := f(x) for all x ∈ S. (2) A pre-monoid is a set M endowed with structure by the prescription of a mapping cmb : M×M → M called combination, which satisfies the associative axiom cmb(cmb(a, b), c) = cmb(a, cmb(b, c)) for all a, b, c ∈ M. (3) Amonoid M is a pre-monoid, with combination cmb, endowed with additional structure by the prescription of a neutral nt ∈ M which satisfies the neutrality axiom cmb(a,nt) = cmb(nt, a) = a for all a ∈ M. (4)

Immediate fixpoints and their use in groundness analysis

Abstract: A theorem by Schroder says that for a certain natural class of functions F: B → B defined on a Boolean lattice B, F(x)=F(F(F(x))) for all x ∃ B. An immediate corollary is that if such a function is monotonic then it is also idempotent, that is, F(x)=F(F(x)). We show how this corollary can be extended to recognize cases where recursive definitions can immediately be replaced by an equivalent closed form, that is, they can be solved without Kleene iteration. Our result applies more generally to distributive lattices. It has applications for example in the abstract interpretation of declarative programs and deductive databases. We exemplify this by showing how to accelerate simple cases of strictness analysis for first-order functional programs and, perhaps more successfully, groundness analysis for logic programs.

Idempotent graphs, weak perfectness, and zero-divisor graphs

Abstract: The idempotent graph I(R) of a ring R is a graph with nontrivial idempotents of R as vertices, and two vertices are adjacent in I(R) if and only if their product is zero. In the present paper, we prove that idempotent graphs are weakly perfect. We characterize the rings whose idempotent graphs have connected complements. As an application, the idempotent graph of an abelian Rickart ring R is used to obtain the zero-divisor graph $$\Gamma (R)$$ of R.

Tilting modules over Auslander-Gorenstein Algebras.

Abstract: For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $\Lambda$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}\Lambda$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $\Lambda$ and its factor algebra $\Gamma=\Lambda/(e)$, we show that there is a bijection between $\tilt_1\Lambda$ and the set $\sttilt\Gamma$ of isomorphism classes of basic support $\tau$-tilting $\Gamma$-modules, where $e$ is an idempotent such that $e\Lambda $ is the additive generator of projective-injective $\Lambda$-modules.