Band of semi-groups

of a given family

A semi-group that has a partition into sub-semi-groups whose (isomorphism) classes are just the semi-groups , and such that for any there is an such that . is also said to be decomposable into the band of semi-groups . In other words, has a partition into a band of semi-groups if all the are sub-semi-groups of and if there is a congruence on such that the -classes are just the . The semi-groups 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 on a semi-group determines a partition of into a band if and only if the quotient semi-group 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 is said to be commutative if for the corresponding congruence the quotient semi-group is commutative; then is a semi-lattice (in this case, is frequently called a semi-lattice of semi-groups ; in particular, if is a chain, then is called a chain of semi-groups ). A band of semi-groups is called rectangular (sometimes matrix) if is a rectangular semi-group (see Idempotents, semi-group of). Equivalently, if the components of the band can be indexed by pairs of indices , where and run over certain sets and , respectively, such that for any one has . 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 there is a finest congruence on any semi-group for which the corresponding quotient semi-group has the property , that is, there exist greatest (or biggest quotient) partitions of 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 and from different components, the product 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 is totally ordered, and for any such that , and for any , one has ; the ordinal sum is defined uniquely up to an isomorphism, by specifying the components and their ordering.

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A congruence on a semi-group is an equivalence relation such that for all one has