Gauss semi-group

A commutative semi-group with unit (i.e. Monoid) satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. non-invertible elements that cannot be represented as a non-trivial product of non-invertible factors); moreover, for each two such decompositions $$ a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l $$ one has $k=l$ and, possibly after renumbering the factors, also $$ b_1 = c_1 \epsilon_1,\ \ldots,\ b_k = c_k \epsilon_k $$

where $\epsilon_1,\ldots,\epsilon_k$ 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.

References

[1]	A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)