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The multiplicative group of a field and hyperidentities. (English. Russian original) Zbl 0705.12002
Math. USSR, Izv. 35, No. 2, 377-391 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1040-1055 (1989).
This paper gives the following characterization of the multiplicative group of a field (Theorem 2): A monoid may be the multiplicative group of a field if and only if there is a regular binary representation of this monoid satisfying the hyperidentitiy of idempotence $$X(x,x)=x$$. A faithful binary representation of a monoid $$M$$ over a set $$Q$$ is an injective homomorphism from $$M$$ to the set of binary operations on $$Q$$ with Mann composition, i.e. $$(A\cdot B)(x,y)=A(x,B(x,y))$$, and such a representation is called regular iff it is right invertible, $$i$$-transitive for $$i=2,3$$ and satisfies the left distributivity hyperidentity. For instance if $$G$$ is the multiplicative group of a field $$Q$$ the corresponding binary representation is given by $$g(x,y)=(1-g)x+gy$$ for $$g\in G$$, and $$x,y\in Q$$.

MSC:
 12E99 General field theory 12L99 Connections between field theory and logic 20M30 Representation of semigroups; actions of semigroups on sets 08B99 Varieties
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