Some Studies On Varieties Of Bands

Let Y be a finite chain with zero,L=[Y;Lα,ρα,β,Φα,β]be a left regular band with neither zero nor identity element.It is proved that L is subdirectly irreducible if and only if1.((?)α∈Y)((?)Nα(?)L0)ρα,0=ωNα∪εL0;2.((?)α,b∈L0)∩α∈Y Nα={a,b};3.((?)α∈Y) Φαa,0is an injection;4.((?)α,β∈Y) α(?)β(?)|Nβ-Nα|=1;5.((?)α∈Y)((?)u,v∈Lα) uΦαa,vΦαb,0=b.The least generating set for each subvarieties of regular bands is given.1.(?)=<L2>;2.(?)=<R2>;3.(?)=<L2×R2>;4.(?)=<Y2>;5.(?)=<L20>;6.(?)=<R20>;7.(?)=<BN>;8.(?)=<L21>;9.(?)=<L21>; 10.(?)=<BL>;11.(?)=<BR>;12.(?)=<B1>.This thesis consists of three chapters.In Chapter1, we introduce some definitions and lemmas, fix some notation which is used in the remaining part.In Chapter2, we give a characterization of left regular subdirectly irreducible band with neither zero nor identity element, in which the structural semilattice is a finite chain.In Chapter3, the least generating set for each subvarieties of regular bands is given.