A commutative semi-group with unit satisfying the cancellation law, in which any non-invertible element is decomposable into a product of irreducible elements (i.e. elements that cannot be represented as a product of non-invertible factors); moreover, for each two such decompositions
one has and, possibly after renumbering the factors, also
where are invertible elements. Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor.
|||A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)|
Gauss semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_semi-group&oldid=15403