Band of semi-groups

of a given family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152101.png" />

A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152102.png" /> that has a partition into sub-semi-groups whose (isomorphism) classes are just the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152103.png" />, and such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152104.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152106.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152107.png" /> is also said to be decomposable into the band of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152108.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b0152109.png" /> has a partition into a band of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521010.png" /> if all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521011.png" /> are sub-semi-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521012.png" /> and if there is a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521014.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521015.png" />-classes are just the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521016.png" />. The semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521017.png" /> are called the components of the given band. The term "band of semi-groups" is consistent with the frequent use of the word "band20M14band" as a synonym of "semi-group all elements of which are idempotents" , since a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521018.png" /> on a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521019.png" /> determines a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521020.png" /> into a band if and only if the quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521021.png" /> is a semi-group of idempotents.

Many semi-groups are decomposable into a band of semi-groups with one or other "better" property; thus, the study of their structure is reduced in some measure to a consideration of the types to which the components of a band belong, and of semi-groups of idempotents (see, e.g. Archimedean semi-group; Completely-simple semi-group; Clifford semi-group; Periodic semi-group; Separable semi-group).

A band of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521022.png" /> is said to be commutative if for the corresponding congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521023.png" /> the quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521024.png" /> is commutative; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521025.png" /> is a semi-lattice (in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521026.png" /> is frequently called a semi-lattice of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521027.png" />; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521028.png" /> is a chain, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521029.png" /> is called a chain of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521030.png" />). A band of semi-groups is called rectangular (sometimes matrix) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521031.png" /> is a rectangular semi-group (see Idempotents, semi-group of). Equivalently, if the components of the band can be indexed by pairs of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521034.png" /> run over certain sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521036.png" />, respectively, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521037.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521038.png" />. Any band of semi-groups is a semi-lattice of rectangular bands, that is, its components can be arranged into subfamilies so that the union of the components of each subfamily is a rectangular band of components, and the original semi-group is decomposable into a semi-lattice of these unions (Clifford's theorem [1]). Since the properties of being a semi-group of idempotents, a semi-lattice or a rectangular semi-group are characterized by identities, for each of the listed properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521039.png" /> there is a finest congruence on any semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521040.png" /> for which the corresponding quotient semi-group has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521041.png" />, that is, there exist greatest (or biggest quotient) partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521042.png" /> into a band of semi-groups, into a commutative band of semi-groups and into a rectangular band of semi-groups.

The term strong band concerns special types of bands of semi-groups [4]: For any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521044.png" /> from different components, the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521045.png" /> is a power of one of these elements. An important special case of a strong band, and also a special case of a chain of semi-groups, is the ordinal sum (or sequentially-annihilating band): The set of its components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521046.png" /> is totally ordered, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521048.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521050.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521051.png" />; the ordinal sum is defined uniquely up to an isomorphism, by specifying the components and their ordering.

References

[1]	A.H. Clifford, "Bands of semi-groups" Proc. Amer. Math. Soc. , 5 (1954) pp. 499–504
[2]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961)
[3]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[4]	L.N. Shevrin, "Strong bands of semi-groups" Izv. Vyssh. Uchebn. Zaved. Mat. : 6 (1965) pp. 156–165 (In Russian)

Comments

A congruence on a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521052.png" /> is an equivalence relation such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521053.png" /> one has

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015210/b01521054.png" />