View source for Principal factor

''of a semi-group''

A Rees quotient semi-group (cf. [[Semi-group|Semi-group]]) of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746802.png" /> is the two-sided [[Principal ideal|principal ideal]] of the semi-group generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746805.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746806.png" />-class (cf. [[Green equivalence relations|Green equivalence relations]]) containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746807.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746808.png" /> is not empty, then it is an ideal, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746809.png" />, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468010.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468011.png" />-simple semi-group or an ideally-simple semi-group (cf. [[Simple semi-group|Simple semi-group]]). The last situation occurs if and only if the semi-group has a kernel (cf. [[Kernel of a semi-group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468012.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468013.png" /> is valid. All regular semi-groups (cf. [[Regular semi-group|Regular semi-group]]) are semi-simple. If each principal factor of a semi-group is either completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468014.png" />-simple or completely simple (cf. [[Completely-simple semi-group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468015.png" />-class the partially ordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468016.png" />-classes (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468017.png" />-classes) contained in it has a minimal element; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468018.png" />.

Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally-simple and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468019.png" />-simple semi-groups in the theory of semi-groups.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>



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