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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 \(x,y\in Q\).

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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