Groupoids assigned to relational systems

Ivan Chajda; Helmut Länger

Displaying similar documents to “Groupoids assigned to relational systems”

Quotients and homomorphisms of relational systems

Ivan Chajda, Helmut Länger (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Relational systems containing one binary relation are investigated. Quotient relational systems are introduced and some of their properties are characterized. Moreover, homomorphisms, strong mappings and cone preserving mappings are introduced and the interplay between these notions is considered. Finally, the connection between directed relational systems and corresponding groupoids is investigated.

Travel groupoids

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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In this paper, by a travel groupoid is meant an ordered pair $(V, *)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u, v \in V$ : $(u * v) * u = u; if (u * v) * v = u, then u = v .$ Let $(V, *)$ be a travel groupoid. It is easy to show that if $x, y \in V$ , then $x * y = y$ if and only if $y * x = x$ . We say that $(V, *)$ is on a (finite or infinite) graph $G$ if $V (G) = V$ and $E (G) = {{u, v} u, v \in V and u \neq u * v = v} .$ Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.

Completely dissociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2012)

Mathematica Bohemica

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In a groupoid, consider arbitrarily parenthesized expressions on the $k$ variables $x_{0}, x_{1}, \dots x_{k - 1}$ where each $x_{i}$ appears once and all variables appear in order of their indices. We call these expressions $k$ -ary formal products, and denote the set containing all of them by $F^{σ} (k)$ . If $u, v \in F^{σ} (k)$ are distinct, the statement that $u$ and $v$ are equal for all values of $x_{0}, x_{1}, \dots x_{k - 1}$ is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative...