Abstract
Abelianity has two different meanings in universal algebra. On the one hand, the term “abelian” is used interchangeably with “commutative” whilst on the other, an algebra is said to be abelian if for every term \({t(x, \overline{y})}\) and for all elements \({a, b, \overline{c}, \overline{d}}\) we have the following implication: \({t(a, \overline{c}) = t(a, \overline{d}) \Rightarrow t(b, \overline{c}) = t(b, \overline{d})}\). These two definitions are equivalent for groups but not generally. We will introduce the class of loosely-abelian algebras which for finite algebras is a generalization of both kinds of abelianity mentioned above. We will prove some basic properties of loosely-abelian algebras and using the introduced concept, we will characterize the subreducts of finite semilattices. Furthermore, we will present an algorithm which solves equations over loosely-abelian algebras.
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Presented by K. Kearnes.
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Krzaczkowski, J. Loosely-abelian algebras. Algebra Univers. 75, 331–350 (2016). https://doi.org/10.1007/s00012-016-0382-3
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DOI: https://doi.org/10.1007/s00012-016-0382-3