Displaying similar documents to “Polynomials in idempotent commutative groupoids”

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 , ⋯ 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 ∈ F σ ( k ) are distinct, the statement that u and v are equal for all values of x 0 , x 1 , ⋯ 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...

Groupoids assigned to relational systems

Ivan Chajda, Helmut Länger (2013)

Mathematica Bohemica

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By a relational system we mean a couple ( A , R ) where A is a set and R is a binary relation on A , i.e. R ⊆ A × A . To every directed relational system 𝒜 = ( A , R ) we assign a groupoid 𝒢 ( 𝒜 ) = ( A , · ) on the same base set where x y = y if and only if ( x , y ) ∈ R . We characterize basic properties of R by means of identities satisfied by 𝒢 ( 𝒜 ) and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.

Two-generated idempotent groupoids with small clones

J. Gałuszka (2001)

Colloquium Mathematicae

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A characterization of all classes of idempotent groupoids having no more than two essentially binary term operations with respect to small finite models is given.

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 ∈ 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 ∈ 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 ∈ V and u ≠ u * v = v } . Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.