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Directoids are introduced by Ivan Chajda.In this paper, a weak directoid is introduced. Some equivalent conditions to a weak directoid to be directoid are derived. Some properties of congruences on the commutative directoid and ortho complemented week directoids are derived.

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

Rodriguez-Villegas conjectured four supercongruences associated to certain elliptic curves, which were first confirmed by Mortenson by using the Gross–Koblitz formula. In this paper we prove four supercongruences between two truncated hypergeometric series$_{2}F_{1}$. The results generalise the four Rodriguez-Villegas supercongruences.

We interpret the support

For the$(d+1)$-dimensional Lie group$G=\\mathbb{Z}_{p}^{\\times }\\ltimes \\mathbb{Z}_{p}^{\\oplus d}$, we determine through the use of$p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian$L$-functions arises from an element in$K_{1}(\\mathbb{Z}_{p}\\unicode[STIX]{x27E6}G\\unicode[STIX]{x27E7})$. If$E$is a semistable elliptic curve over$\\mathbb{Q}$, these abelian$L$-functions already exist; therefore, one can obtain many new families of higher order$p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

Let Apéry numbers An'=∑k=0n(Cnk)2Cn+kk. We proved a congruence involving An' modulo p4 by using some unknown congruences involving harmonic numbers and Bernoulli numbers, and an identity involving sums of binomial coefficients by Zeilberger’s algorithm.

In this paper, we give the generic classification of the singularities of 3-parameter line congruences in$\\mathbb {R}^{4}$. We also classify the generic singularities of normal and Blaschke (affine) normal congruences.

For a measurable space (𝑋,𝒜), let 𝑀⁺(𝑋,𝒜) be the commutative semiring of non-negative realvalued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of 𝑀⁺(𝑋,𝒜) and the ideal lattice of its ring of differences 𝑀(𝑋,𝒜). Moreover, we infer that each ideal of 𝑀⁺(𝑋,𝒜) is a semiring 𝑧-ideal. We investigate the duality between cancellative congruences on 𝑀⁺(𝑋,𝒜) and 𝑍𝒜-filters on 𝑋. We observe that every 𝜎-algebra is a completely regular 𝜎-frame, so compactness and pseudocompactness coincide in 𝜎-algebras, and we provide a new characterization for compact measurable spaces via algebraic properties of 𝑀⁺(𝑋,𝒜). It is shown that the space of (real) maximal congruences on 𝑀⁺(𝑋,𝒜) is homeomorphic to the space of (real) maximal ideals of the 𝑀(𝑋,𝒜). We solve the isomorphism problem for the semirings of the form 𝑀⁺(𝑋,𝒜) for compact and realcompact measurable spaces.

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

In this article, the notions of admissible congruences and normal congruences on a weakly type B semigroup are characterized and the relationship between admissible congruences and normal congruences is investigated. In particular, some properties of such congruences on a weakly type B semigroup are given using an approach of kernel-trace. Finally, we extend the congruence pair on an inverse semigroup to the case of a weakly type B semigroup and obtain some results.