Proof of a Lin-Peng-Toh's conjecture on an Andrews-Beck type congruence

Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities.     

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     

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.     

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.     

Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In Sánchez Terraf and Vaggione (Trans Am Math Soc, in press) it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general. Supported by Conicet.     

Local Connectedness Made Uniform

The uniformly locally connected reflection for a locally connected uniform frame is constructed. Applications of this construction to the theory of locally connected completely regular frames are given. One such application is that if a completely regular frame is locally connected and pseudocompact then every compactification of it is locally connected.     

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.     

关于同余式Σsum from j=1 to n ajxjdj≡b（mod pi）的解的个数

设N_p^l代表同余式Σsum from j=1 to n a_jx_j^d_j≡b（mod pⁱ）的解的个数，这里ｐ是一个奇素数,p|ba₁…a_n,d_j|p-1,d_j>1,j=1,…,n.本文给出N_p(b)一个渐近公式．     

∑_(i=1)~n x_i／d_i表示的最小整数及其应用

设I(d1,…,dn)为方程∑i=1n xi/di≡0(mod 1),1≤xi≤di-1的解的个数．若I(d1,…,dn)>0,令L(d1,…,dn)为∑i=1n xi/di(1≤xi≤di-1)表示的最小整数．I(d1,…,dn),L(d1,…,dn)及它们的估计在有限域上对角方程解数的研究中具有重要作用．本文给出了等式成立的若干充分条件,这里ωi=gcd(di,lcm[dj:j≠i]),i=1,…,n．