Maximal antichains of minimum size

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose size is in $K$, and $A\not\subseteq B$ for all ${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for the case $K={2,4}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference

On the number of minimal completely separating systems and antichains in a Boolean lattice

An (n)completely separating system C ((n)CSS) is a collection of blocks of [n] = {1,..., n} such that for all distinct a, b ∈ [n] there are blocks A, B ∈C with a ∈ A \ B and b ∈ B \ A. An (n)CSS is minimal if it contains the minimum possible number of blocks for a CSS on [n]. The number of non-isomorphic minimal (n)CSSs is determined for 11 ≤ n ≤ 35. This also provides an enumeration of a natural class of antichains

Lattices of congruence relations for inverse semigroups

The study of congruence relations is acknowledged as fundamental to the study of algebras. Inverse semigroups are a widely studied class for which congruences are well understood. We study one sided congruences on inverse semigroups. We develop the notion of an inverse kernel and show that a left congruence is determined by its trace and inverse kernel. Our strategy identifies the lattice of left congruences as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. This is a natural way to describe one sided congruences with many desirable properties, including that a pair is the inverse kernel and trace of a left congruence precisely when it is the inverse kernel and trace of a right congruence. We classify inverse semigroups for which every Rees left congruence is finitely generated, and provide alternative proofs to classical results, including classifications of left Noetherian inverse semigroups, and Clifford semigroups for which the lattice of left congruences is modular or distributive. In the second half of this thesis we study the partial automorphism monoid of a finite rank free group action. Congruences are described in terms a Rees congruence, subgroups of direct powers of the group and a subgroup of the wreath product of the group and a symmetric group. Via analysis of the subgroups arising in this description we show that, for finite groups, the number of congruences grows polynomially in the rank of action with an exponent related to the chief length of the group

Quantum Gravity: From continuous to discrete

The consistent definition of a quantum gravity theory has to overcome several obstacles. Here we take important steps in the development of three approaches to quantum gravity. By utilising matter fields as mediators from ultraviolet to infrared energies, we study a coupling between asymptotically-safe quantum gravity and the hypercharge. The resulting symmetry enhancement allows for a possible ultraviolet completion of the joined system, predicting the infrared value of the hypercharge within estimated systematic errors, thereby increasing the predictive power of the model. Additionally, previous studies suggest that K ̈ahler-Dirac fermions on Euclidean dynamical triangulations do not spontaneously break chiral symmetry. Here we develop computational tools accounting for the back reaction of fermions on the lattice. If extended studies support the evidence that chiral symmetry remains intact, then the model passes an important observational viability test. Lastly, we provide procedures allowing for the extraction of geometrical and topological properties from a causal set. Specifically, we build a spatial distance function which can be used to construct dimensional estimators for the Hausdorff and spectral dimension. In agreement with other quantum-gravity approaches, the latter exhibits a form of dimensional reduction at high energies on account of the inherent non-localness of causal sets

Achieving while maintaining:A logic of knowing how with intermediate constraints

In this paper, we propose a ternary knowing how operator to express that the agent knows how to achieve $\phi$ given $\psi$ while maintaining $\chi$ in-between. It generalizes the logic of goal-directed knowing how proposed by Yanjing Wang 2015 'A logic of knowing how'. We give a sound and complete axiomatization of this logic.Comment: appear in Proceedings of ICLA 201

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum