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Free ringoids and imbeddings of ringoids into rings. (Russian) Zbl 0725.20047

By a ringoid we shall mean an algebra (K,\(\alpha\),\(\circ)\), where ar \(\alpha\) \(=3\), ar \(\circ =2\) and \(\alpha (\alpha (x,y,z),s,t)=\alpha (x,y,\alpha (z,s,t))\), \(\alpha (x,y,y)=x\), \(\alpha (x,x,y)=y\), \(\alpha (x,y,z)=\alpha (z,y,x)\), \(x\circ (y\circ z)=(x\circ y)\circ z\), \(x\circ \alpha (y,z,t)=\alpha (x\circ y,x\circ z,x\circ t)\), \(\alpha (x,y,z)\circ t=\alpha (x\circ t,y\circ t,z\circ t)\). The author considers the construction of free ringoids. He shows that every ringoid can be embedded into a ringoid with zero (a ringoid with unit, a ringoid of endomorphisms of an abelian heap).

MSC:

20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
17A40 Ternary compositions
17A50 Free nonassociative algebras
08A62 Finitary algebras
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