A CAYLEY THEOREM FOR THE MULTIPLICATIVE SEMIGROUP OF A FIELD

Since there exist two commutative elementarily equivalent semigroups of which one is the multiplicative semigroup of a field and the other is not a multiplicative semigroup of any field, it is impossible to characterize multiplicative semigroups of fields by formulas of the first order language (logic). In this work we characterize the multiplicative semigroup of a field by its binary representation (Cayley type theorem).

Some properties of linear right ideal nearrings

In a previous paper, we determined all those topological nearrings𝒩nwhose additive groups are then-dimensional Euclidean groups,n>1, and which containnone-dimensional linear subspaces{Ji}i=1nwhich are also right ideals of the nearring with the property that for eachw∈𝒩n, there existwi∈Ji,1≤i≤n, such thatw=w1+w2+⋯+wnandvw=vwnfor eachv∈𝒩n. In this paper, we determine the properties of these nearrings, their ideals, and when two of these nearrings are isomorphic, and we investigate the multiplicative semigroups of these nearrings.