The Congruence Lattice of Bruck–Reilly Extensions

We study the lattice. C(S) of congruences of a monoid S which is the Bruck-Reilly extension of a monoid T by a homomorphism . The inclusion, meet and join of congruences are described in terms of congruences and ideals of T. We show that C(S) can be naturally decomposed into three sublattices, corresponding (roughly speaking) to the three different types of congruences on such semigroups.1991 Mathematics Subject Classification: 20M10     

A new proof of the Ramanujan congruences for the partition function

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

正则半群的局部化及应用

证明了正则半群S在其幂等元素E（S）所生成的子半群＜E（S）＞上的局部化在同构意义下存在惟一，其为其最大群同态象，同时也给出了其最小群同余。     

On the behavior of some two-variable p-adic L-functions

We give some p-adic integral representations for the two-variable p-adic L-functions introduced recently by G. Fox. For powers of the Teichmüller character, we use the integral representation to extend the L-function to a larger domain, in which it is a meromorphic function in the first variable and an analytic element in the second. These integral representations imply systems of congruences for the generalized Bernoulli polynomials, improving previous results of Fox, Gunaratne, and the author; they also lead to generalizations of some formulas of Diamond and of Ferrero and Greenberg for p-adic L-functions in terms of the p-adic gamma and log gamma functions.     

NOTES ON GLAISHER''''S CONGRUENCES

Let p be an odd prime and let n ≥ 1,k ≥ 0 and r be integers. Denote by Bk the kth Bernoulli number. It is proved that (i) If r≥1 is odd and suppose p≥r+4, thenp-1∑j=11/(np+j)=-(2n+1+(x+1)/2(x+2)Bp-r-2p2(modp3).(ii)IFr≥2is even and suppose p≥r+3,thenp-1∑j=11/(np+j)+=r/r+1Bp-x-1p(modp2).(iii)p-1∑j=11/(np+j)p-2=-(2n+1)p(modp2).Thisesult generalizes the Glaisher's congruence. As a corollary, a generalization of the Wolstenholme's theorem is obtained.     

$ {\cal L} $-subvarieties of the variety of idempotent semirings

We show that certain varieties of idempotent semirings are determined by some properties of Green's relations, provide equational bases for them and give conditions guaranteeing that some Green's relations are congruences. Received November 2, 1999; accepted in final form May 2, 2000.     

Congruences on eventually regular semigroups

Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored.