# Gauss semi-group

A commutative semi-group with unit (i.e. Monoid) satisfying the cancellation law and 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.