Semigroups Decomposable into Rectangular Bands
The concept of a band is one of the main tools of the structural theory of semigroups. Therefore the question is of interest how various decompositions into bands behave under lattice isomorphisms of semigroups. We have touched upon this question in Section 35 in connection with the decomposition of commutative semigroups into a semilattice of archimedean components. Decompositions of [sub]semilattices into an ordinal sum were the main tool in the considerations of Section 36. In the present chapter, as is seen from the title, the decompositions into rectangular bands will be the object of attention. We recall (see Theorem 1.7.1) that an arbitrary band of some family of semigroups is a semilattice of rectangular bands of these semigroups divided into subfamilies. So, from the point of view of decompositions into arbitrary bands, rectangular bands are of particular interest.
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