Ideal series

of a semi-group $S$

A sequence of sub-semi-groups

$$A_1\subset A_2\subset\dotsb\subset A_m=S\label{*}\tag{*}$$

such that $A_i$ is a (two-sided) ideal of $A_{i+1}$, $i=1,\dotsc,m-1$. The sub-semi-group $A_1$ and the Rees factor semi-groups $A_{i+1}/A_i$ (see Semi-group) are called the factors of the series \eqref{*}. Two ideal series are said to be isomorphic if a one-to-one correspondence can be established between the factors such that corresponding factors are isomorphic. An ideal series

$$B_1\subset B_2\subset\dotsb\subset B_n=S$$

is said to be a refinement of \eqref{*} if every $A_i$ occurs among the $B_j$. An ideal series is a composition series if it does not have proper refinements. Any two ideal series of a semi-group have isomorphic refinements; in particular, in a semi-group having a composition series all such series are isomorphic (the analogue of the theorems of Schreier and Jordan–Hölder for normal series in groups, see [1], [2]). An ideal series is a chief series if its terms are ideals in the whole semi-group and if it has no proper refinements consisting of ideals of the semi-group. If a semi-group has a composition series, then it also has a chief series; the converse is false. In a semi-group with a chief series, its factors are isomorphic to the chief factors (cf. Principal factor) of $S$.

As for normal series in groups, the concepts mentioned above (as well as their properties) naturally generalize to the case of infinite systems of nested sub-semi-groups. In particular, an ascending ideal series in a semi-group $S$ is a totally ordered sequence

$$A_1\subset\dotsb\subset A_\alpha\subset A_{\alpha+1}\subset\dotsb\subset A_\beta=S,$$

where at limit points there stand the unions of the preceding members, and $A_\alpha$ is an ideal of $A_{\alpha+1}$ for all $\alpha<\beta$.

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)