Abstract
Let be a set and the set of all binary operations on . We say that is a distributive set of operations if all pairs of elements are right distributive, that is, (we allow ).
The question of which groups can be realized as distributive sets was asked by J. Przytycki. The initial guess that embedding into for some holds for any was complicated by an observation that if is idempotent (), then commutes with every element of . The first noncommutative subgroup of (the group ) was found in October 2011 by Y. Berman.
Here we show that any group can be embedded in for (as a set). We also discuss minimality of embeddings observing, in particular, that with six elements is the smallest set such that contains a nonabelian subgroup.
Citation
Gregory Mezera. "Embedding groups into distributive subsets of the monoid of binary operations." Involve 8 (3) 433 - 437, 2015. https://doi.org/10.2140/involve.2015.8.433
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