On Birkhoff's quasigroup axioms. (Q279714)
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scientific article; zbMATH DE number 6575136
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On Birkhoff's quasigroup axioms. |
scientific article; zbMATH DE number 6575136 |
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On Birkhoff's quasigroup axioms. (English)
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29 April 2016
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left quasigroups
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quasigroup axioms
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Birkhoff identities
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varieties of quasigroups
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division groupoids
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cancellation groupoids
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Quasigroups form a variety only if two additional operations, namely \(/\) and \(\backslash\) are considered. Birkhoff formulated six identities that define equationally the variety of quasigroups. Evans proved that only the most natural four identities are needed.NEWLINENEWLINE In this article the authors study which other four-tuples of Birkhoff's identities suffice to define quasigroups. Among all combinations, nine define quasigroups, four define larger classes and two remain open.
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