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A duality between infinitary varieties and algebraic theories

Jiří Adámek, Václav Koubek, Jiří Velebil (2000)

Commentationes Mathematicae Universitatis Carolinae

A duality between λ -ary varieties and λ -ary algebraic theories is proved as a direct generalization of the finitary case studied by the first author, F.W. Lawvere and J. Rosick’y. We also prove that for every uncountable cardinal λ , whenever λ -small products commute with 𝒟 -colimits in Set , then 𝒟 must be a λ -filtered category. We nevertheless introduce the concept of λ -sifted colimits so that morphisms between λ -ary varieties (defined to be λ -ary, regular right adjoints) are precisely the functors...

An algebraic version of the Cantor-Bernstein-Schröder theorem

Hector Freytes (2004)

Czechoslovak Mathematical Journal

The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to σ -complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for...

Completely dissociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2012)

Mathematica Bohemica

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 law holds...

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