Internal monoids and groups in the category of commutative cancellative medial magmas. (English) Zbl 1358.08002
Groupoids, i.e., algebras \((A,\oplus )\) with one binary operation \(\oplus\), are considered under the name “magmas”. If \(\oplus\) is commutative, cancellative and medial (the last one means \((x\oplus y)\oplus (z\oplus w)=(x\oplus z)\oplus (y\oplus w)\)) the magma \((A,\oplus )\) is called a ccm-magma. Several examples of ccm-magmas are presented. The authors study the category of ccm-magmas. This category is weakly Mal’tsev. The existence of an internal monoid structure \(A\times A \to A \leftarrow 1\) is characterized for a given object \((A,\oplus )\) and for every choice of a unit element. Some properties of internal relations are studied.
Reviewer: G. I. Zhitomirskij (Herzliyya)
MSC:
08C05 | Categories of algebras |
20N02 | Sets with a single binary operation (groupoids) |
08C15 | Quasivarieties |
Keywords:
midpoint algebra; commutative monoid; abelian group; cancellation law; medial law; quasigroup; internal monoid; internal group; internal relation; weakly Mal’tsev categoryReferences:
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