Principal factor

of a semi-group

A Rees quotient semi-group (cf. Semi-group) of the form $J(x)/N(x)$, where $J(x)$ is the two-sided principal ideal of the semi-group generated by the element $x$ and $N(x)=J(x)\setminus J_x$, where $J_x$ is the $\mathcal J$-class (cf. Green equivalence relations) containing $x$. If the set $N(x)$ is not empty, then it is an ideal, and if $N(x)=\emptyset$, one puts $J(x)/N(x)=J(x)$. A principal factor of a semi-group is also known as an ideal factor. An arbitrary principal factor of a semi-group is either a semi-group with zero multiplication, a $0$-simple semi-group or an ideally-simple semi-group (cf. Simple semi-group). The last situation occurs if and only if the semi-group has a kernel (cf. Kernel of a semi-group) and this kernel coincides with the given principal factor. A semi-group without a principal factor with zero multiplication is said to be semi-simple; the condition of semi-simplicity of a semi-group is equivalent, for example, to the fact that for any of its two-sided ideals $A$ the equality $A^2=A$ is valid. All regular semi-groups (cf. Regular semi-group) are semi-simple. If each principal factor of a semi-group is either completely $0$-simple or completely simple (cf. Completely-simple semi-group), then the semi-group is called completely semi-simple. A semi-group is completely semi-simple if and only if it is regular and satisfies any of the following (mutually-dual) conditions: for each $\mathcal J$-class the partially ordered set of $\mathcal L$-classes (or $\mathcal R$-classes) contained in it has a minimal element; in this case $\mathcal J=\mathcal D$.

Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally-simple and $0$-simple semi-groups in the theory of semi-groups.

References

[1]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)

Comments

A principal factor of a semi-group is also called a chief factor of a semi-group.