## 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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