We investigate semigroups S which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on S. We call such an S a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups, including periodic, stable, inverse and several well-known types of simple semigroups, the only DSC members are groups. However, it turns out that there exist nongroup DSC semigroups, which we obtain by utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.

A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair $(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n, each term of which is congruent to P modulo $2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If $3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout’s identity.

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.

Do policy priorities that candidates emphasize during election campaigns predict their subsequent legislative activities? We study this question by assembling novel data on legislative leadership posts held by Japanese politicians and using a fine-tuned transformer-based machine learning model to classify policy areas in over 46,900 statements from 1270 candidate manifestos across five elections. We find that a higher emphasis on a policy issue increases the probability of securing a legislative post in the same area. This relationship remains consistent across multiple elections and persists even when accounting for candidates' previous legislative leadership roles. We also discover greater congruence in distributive policy areas. Our findings indicate that campaigns provide meaningful signals of policy priorities.

A notion of normal submonoid of a monoid M is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf {NorSub}(M)$ of normal submonoids of M is a complete lattice. Joins are explicitly described and the lattice is computed for the finite full transformation monoids $T_n$, $n\geq ~1$. It is also shown that $\mathsf {NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that it, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf {Cong}(M)$ of congruences on M. This leads to a new strategy for computing $\mathsf {Cong}(M)$ consisting of computing $\mathsf {NorSub}(M)$ and the so-called unital congruences on the quotients of M modulo its normal submonoids. This provides a new perspective on Malcev’s computation of the congruences on $T_n$.

While previous research has identified the performance implications of leaders’ positive implicit followership theories (IFTs, i.e., personal expectations regarding followers’ positive characteristics), this study focuses on the effect of leader–follower congruence in positive IFTs on followers’ job performance. To test our predictions, we conducted two complementary studies. The results of Study 1 (an experiment, N = 200) show that leader–follower congruence (versus incongruence) in positive IFTs is positively related to followers’ relational identification with the leader, which, in turn, is positively related to followers’ job performance. Moreover, followers’ uncertainty avoidance strengthens this relationship. These findings were replicated in Study 2 (a three-wave survey, N = 223) through polynomial regression and response surface analysis. This study improves our understanding of IFTs by showing that leader–follower congruence in this domain is related to followers’ outcomes.

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv _w$ introduced by Šobot is an equivalence relation on $\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv _w$ coincides with the strong congruence relation $\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient $(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv _w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat {{\mathbb Z}}$ of the integers.

Several recent studies have found unequal policy responsiveness, meaning that the policy preferences of high-income citizens are better reflected in implemented policies than the policy preferences of low-income citizens. This has been found mainly in a few studies from the US and a small number of single-country studies from Western Europe. However, there is a lack of comparative studies that stake out the terrain across a broader group of countries. We analyze survey data on the policy preferences of about 3,000 policy proposals from thirty European countries over nearly forty years, combined with information on whether each policy proposal was implemented or not. The results from the cross-country data confirm the general pattern from previous studies that policies supported by the rich are more likely to be implemented than those supported by the poor. We also test four explanations commonly found in the literature: whether unequal responsiveness is exacerbated by (a) high economic inequality, (b) the absence of campaign finance regulations, (c) low union density, and (d) low voter turnout.

A semigroup S is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair $(s,t)\in S\times S$ as a consequence of those in U. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids, and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green’s preorders ${\leq _{\mathcal {L}}}$ or ${\leq _{\mathcal {J}}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or ${\mathcal {J}}$-triviality.

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

Civil society organizations (CSOs) can facilitate collective action. This makes understanding what shapes whether people are likely to engage with CSOs critically important. This paper argues that whether an organization is perceived as congruent – similar to an individual in values – is a key determinant of whether individuals will engage with it. I use a conjoint survey experiment to test how organizational attributes signaling congruence influence respondents’ willingness to attend a hypothetical organization’s meetings. I find that individuals are more likely to choose organizations that are more likely to be congruent with them, except when it comes to funding. These findings imply that an individual’s level of comfort with a CSO matters for engagement; thus, CSOs need to consider how they match to their publics when reaching out to potential joiners. Furthermore, donors seeking to support CSOs need to pay attention to their impact on perceptions of congruence.