Binary operations

DOAJ (DOAJ: Directory of Open Access Journals), 2013

Given a continuous monadic functor T : Comp → Comp in the category of compacta and a discrete topological semigroup X we extend the semigroup operation ϕ : X × X → X to a right-topological semigroup operation Φ : T βX × T βX → T βX whose topological center ΛΦ contains the dense subsemigroup T f X consisting of elements a ∈ T βX that have finite support in X.

For any binary operation, four alternatives exist. It could be (i) both commutative and associative;(ii) neither commutative nor associative;(iii) commutative but not associative;(iv) associative but not commutative.The basic arithmetical operations provide examples for the first two possibilities. This paper presents elementary examples for the last two possibilities. It is claimed that the study of these examples is extremely important in order to understand the logical independence of commutativity and associativity.

Logic and Foundations of Mathematics, 1999

2012

1. Every morphism f is associated with two objects (which may be the same) called the domain and codomain of f. One can view a morphism as an arrow from one object to another thus forming a directed graph. We sometimes write cod(f) and dom(f) to denote these objects, more often we give them names like A and B. We write f : A −→ B or A f −−→ B. ... 2. Given A f −−→ B and B g −−→ C we say f and g are composable (ie cod(f) = dom(g)). We define a binary operation, written ◦, on composable maps so that g ◦ f is a morphism: dom(g ◦ f) = dom(f) and cod(g ◦ f) = ...

Theory and Practice of Logic Programming, 2002

It is important that practical data-flow analyzers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the soundness of Sharing assume the occurs-check. This paper provides a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check. Results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven. automatica di programmi mediante interpretazione astratta." thus we have also S ∈ amgu(sh, y → t).

1990

Fuzzy Sets and Systems, 2004

In this paper, a new approach to the study of binary operations is shown. After introducing the concept of external range, some results concerning the class of continuous binary operations with ÿnite external range are given.

2022

The purpose of this article is to introduce \textit{the geometry of operations}. Let $\psi$ be a bijection from a nonempty set $G$ into another set $G^{'}$, we prove that for any fixed operation on $G$, there exists a \textit{unique} operation on $G^{'}$, where $\psi$ is an \textit{isomorphism}. This finding allow us to show that we can make the set of natural numbers a \textit{commutative field}. Therefore we establish that the symmetric group $\mathcal{S}_{G}$ of a given nonempty set $G$ \textit{acts} on the set of all operations on $G$. Then we generalize the notions of $t$-norms, $t$-conorms and we give the equivalent of the $t$-conorm for \textit{the triangle function}.

FUDMA JOURNAL OF SCIENCES

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

2008

We propose a rule format that guarantees associativity of binary operators with respect to all notions of behavioral equivalence that are defined in terms of (im)possibility of transitions, e.g., the notions below strong bisimilarity in van Glabbeek’s spectrum. The initial format is a subset of the De Simone format. We show that all trivial generalizations of our format are bound for failure. We further extend the format in a few directions and illustrate its application to several formalisms in the literature. A subset of the format is studied to obtain associativity with respect to graph isomorphism.