A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal {S}$ is a quotient of a numerical semigroup with k generators, we call $\mathcal {S}$ a k-quotient. We give a necessary condition for a given numerical semigroup $\mathcal {S}$ to be a k-quotient and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a k-quotient. We also examine the probability that a randomly selected numerical semigroup with k generators is a k-quotient.

We study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.

We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

We study unitary quotients of the free product unitary pivotal category ${{A}_{2}}\,*\,{{T}_{2}}$. We show that such quotients are parametrized by an integer $n\,\ge \,1$ and an $2n$–th root of unity $\omega $. We show that for $n\,=\,1,\,2,\,3$, there is exactly one quotient and $\omega \,=\,1$. For $4\,\le \,n\,\le \,10$, we show that there are no such quotients. Our methods also apply to quotients of ${{T}_{2}}\,*\,{{T}_{2}}$, where we have a similar result.

The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of ${{A}_{2}}\,*\,{{T}_{2}}$ and ${{T}_{2}}\,*\,{{T}_{2}}$, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.

During the preparation of this manuscript, we learnt of Liu's independent result on composites of ${{A}_{3}}$ and ${{A}_{4}}$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch–Haagerup showed that the principal graph of a composite of ${{A}_{3}}$ and ${{A}_{4}}$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\,\ge \,4$.

This is an abridged version of arxiv:1308.5723.

We show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

Let {We}e∈ω be a standard enumeration of the recursively enumerable (r. e.) subsets of ω = {0, 1, 2, …}. The lattice of recursively enumerable sets, is the structure ({We}e∈ω, ∪, ∩). is the sublattice of consisting of the recursive sets.

Suppose is a lattice of subsets of ω. ≡ is said to be a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V ∈ , if U ≡ U′ and V ≡ V′ then U ∪ U′ ≡ V ∪ V′ and U ∩ U′ ≡ V ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V].

The quotient lattices of (or of some sublattice ) correspond naturally with the congruence relations which give rise to them, and in turn the congruence relations of sublattices of can be characterized in part by their computational complexity. The aim of the present paper is to characterize congruence relations in some of the most important complexity classes.

Let {We}e∈ω be a standard enumeration of the recursively enumerable (r.e.) subsets of ω = {0,1,2,…}. The lattice of recursively enumerable sets, , is the structure ({We}e∈ω,∪,∩). ≡ is a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V′ ∈ , if U ≡ U′ and V ≡ V′, then U ∪ V ≡ U′ ∪ V′ and U ∩ V ≡ U′ ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V]. We say that a congruence relation ≡ on is if {(i, j)| Wi ≡ Wj} is . Define =* by putting Wi, =* Wj if and only if (Wi − Wj)∪ (Wj − Wi) is finite. Then =* is a congruence relation. If D is any set, then we can define a congruence relation by putting Wi Wj if and only if Wi ∩ D =* Wj ∩D. By Hammond [2], a congruence relation ≡ ⊇ =* is if and only if ≡ is equal to for some set D.

The Friedberg splitting theorem [1] asserts that if A is any recursively enumerable set, then there exist disjoint recursively enumerable sets A0 and A1 such that A = A0∪ A1 and such that for any recursively enumerable set B

In this paper we give some properties of the pairwise perfectly normal spaces defined by Lane. In particular we prove that a space (X, P, Q) is pairwise perfectly normal if and only if every P(Q)–closed set is the zero of a P(Q)–l.s.c. and Q(P)–u.s.c. function. Also we characterize the pairwise perfect normality in terms of sequences of semicontinuous functions by means of a result which contains the known Tong's characterization of perfectly normal topological spaces, whose proof we modify by using the technique of binary relations.