Some Studies Of Several Classes Of Semigroups Based On Structural Analysis

It is well known that matrix algebra,conformal mapping,wavelets transform,Fourier transform and Laplace transform have been widely used in civil engineering and engineering mechanics.In fact,as far as the essence of mathematics is concerned,conformal transformation,wavelet transform,Fourier transform and Laplace transform all belong to the category of semigroups in mathematics,while matrix algebra is actually a matrix semigroup.Therefore,it is meaningful to study semigroup theory and its application in civil engineering and engineering mechanics based on mathematical theory.The main contents of this thesis are the algebraic theory of several kinds of generalized regular semigroups and their algebraic structures.(1)As a generalization of left inverse semigroups in the class of regular semigroups,a class of semigroups which named (?)-inverse semigroups is introduced.By introducing the concept of left cycle product of semigroups,an algebraic structure of such semigroups is established.It is proved that a semigroup is an (?)-inverse semigroup if and only if it can be expressed as a left cycle product of an E-ample semigroup and a left regular band.This result generalizes a structure theorem of left inverse semigroups by M.Yamada,a famous semigroup expert.(2)The (?)-inverse semigroups is defined.A U-abundant semigroup is called (?)-inverse semigroup,if S satisfies PC condition and characteristic elements form a regular band.By using the circle product of a left regular band,a right regular band and an E-ample semigroup,a structure theorem of (?)-inverse semigroups is given.It is proved that a semigroup S is a (?)-inverse semigroup if and only if S is a circle product of a left regular band,a right regular band and an E-ample semigroup.(3)A superabundant semigroup is called a left regular cyber-group if its idempotents set forms a left regular band.After the investigation of the properties of superabundant semigroups,a structure theorem for the left regular cyber-groups is established by using the newly defined left twist product of semigroups.It is proved that a semigroup S is a left regular cyber-group if and only if S is a left twist product of a left regular band and a C-a semigroup.Moreover,a special example of this result is the structure theorem of left regular Orthogroup semigroups given by famous semigroup expert M.Petrich.(4)Based on the generalized Green relation,the concept of weakly rpp semigroups is considered.Some kind of weakly rpp semigroups,namely,weakly left C-rpp semigroups,is studied.By means of a semilattice decomposition of such semigroups,it is proved that every weakly left C-rpp semigroup can be expressed as a strong Semilattice of a nilpotent monoid and a left cross product of a left regular band.This result is a generalization of J.B.Fountain's theorem on the structure of left C-rpp semigroups and Guo-Shum-Zhu's theorem on left C-rpp semigroups.(5)L-regular semigroups are defined and their properties are discussed.It is shown that a semigroup S is a weakly L-regular semigroup with left central idempotents if and only if S is a strong semilattice of the direct product of a H-left cancellative monoid and a right zero band.Then,it is proved that such kind of semigroups can be described by the strong spined product of a weakly L-regular semigroup with central idempotents and a right normal band.(6)A semigroup S is a U-superaundant semigroup if and only if S is a semilattice of completely J-simple semigroups.Based on the fact,a structure theorem for U-superabundant semigroupsis is obtained in terms of the sets of normalized Rees matrix semigroups over some monoids and their structure mappings.(7)A U-superaundant semigroup is called U-ortho-superaundant semigroup if its distinguish idempotents set forms a band.It is proved that a semigroup S is a U-ortho-superaundant semigroup if and only if S is a semilattice of some kind of generalized rectangular monoids.On this basis,the algebraic structure of U-ortho-superaundant semigroups is described by structure mappings.This result generalizes M.Pretrich's results on Orthodox groups.