Vol. 8, No. 3, 2015

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Embedding groups into distributive subsets of the monoid of binary operations

Gregory Mezera

Vol. 8 (2015), No. 3, 433–437

Let X be a set and Bin(X) the set of all binary operations on X. We say that S Bin(X) is a distributive set of operations if all pairs of elements α,β S are right distributive, that is, (a αb) βc = (a βc) α(b βc) (we allow α = β).

The question of which groups can be realized as distributive sets was asked by J. Przytycki. The initial guess that embedding into Bin(X) for some X holds for any G was complicated by an observation that if S is idempotent (a a = a), then commutes with every element of S. The first noncommutative subgroup of  Bin(X) (the group S3) was found in October 2011 by Y. Berman.

Here we show that any group can be embedded in Bin(X) for X = G (as a set). We also discuss minimality of embeddings observing, in particular, that X with six elements is the smallest set such that Bin(X) contains a nonabelian subgroup.

monoid of binary operations, distributive set, shelf, multishelf, distributive homology, embedding, group
Mathematical Subject Classification 2010
Primary: 55N35
Secondary: 18G60, 57M25
Received: 31 October 2012
Revised: 20 March 2014
Accepted: 27 April 2014
Published: 5 June 2015

Communicated by Kenneth S. Berenhaut
Gregory Mezera
Department of Mathematics
George Washington University
Washington, DC 20052
United States