Two Studies On Subdirectly Irreducible Bands

We first characterize the subdirectly irreducible structure of a band whose struc-tural semilattice has height 2 by using the fundmental semilattice of semigroups.We give a sufficient and necessary condition for this kind of bands to be subdirectly ir-reducible.The second aim is to classify subdirectly irreducible bands with elements less than 9.Given a subdirectly irreducible band with neither identity nor zero,we find that the elements in the lowest D-class B? are at least four,and the elements in D-classes which cover B? are not less than 2.Restricting heights of the structural semilattice and discussing respectively from regularity or irregularity of the band,we prove that,up to isomorphism,the number of subdirectly irreducible bands with elements less than 9 is exactly 51This main part is divided into three chaptersIn Chapter 1,we introduce some definitions and lemmas,fix some notation which is used in the remaining partsIn Chapter 2,the fundamental semilattice of semigroups is used to describe sub-directly irreducible bands whose structural semilattices have height 2.It is proved that the irreducibility of the subdirectly irreducible band is mostly reflected by the properties of two particular elements of B?In Chapter 3,based on the conclusions in the Chapter 2,we give a classification of subdirectly irreducible bands whose cardinality is less than 9.We prove that,up to isomorphism,there exactly exist 51 such bands.