arXiv subjects
Publications and source records attributed to Rob Egrot.
We investigate connections between the free lattice generated by a poset while preserving certain bounds and the canonical extension of a poset. Explicitly, we describe how the free lattice generated by a poset while preserving certain bounds can be constructed as a colimit of `intermediate structures' as they occur in the construction of a canonical extension of a poset.
Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.
We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number $n$ such that $\forall$ always has a winning strategy in the game with $n$ colours whenever $G\not\cong H$. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with $n=3$ for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any $k<ω$) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the $k$-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.
We show that if for all finite $c$ there is a pair of non-isomorphic finite digraphs satisfying some additional conditions, one of which is that they cannot be distinguished in a certain $c$-colour node colouring game, then there can be no axiomatisation of the class of representable relation algebras in any first-order theory of arbitrary quantifier-depth using only finitely many variables. This corrects the proposed strategy of Hirsch and Hodkinson, \emph{Relation algebras by games}, North-Holland (2002), Problem 1. However, even for $c=2$, no pair of non-isomorphic graphs indistinguishable in the game is currently known.
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.
We define an order polarity to be a polarity $(X,Y,R)$ where $X$ and $Y$ are partially ordered, and we define an extension polarity to be a triple $(e_X,e_Y,R)$ such that $e_X:P\to X$ and $e_Y:P\to Y$ are poset extensions and $(X,Y,R)$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a pre-order structure on $X\cup Y$ such that the natural embeddings, $ι_X$ and $ι_Y$, of $X$ and $Y$, respectively, into $X\cup Y$ preserve the order structures of $X$ and $Y$ in increasingly strict ways. We define a Galois polarity to be an extension polarity where $e_X$ and $e_Y$ are meet- and join-extensions respectively, and we show that for such polarities there is a unique pre-order on $X\cup Y$ such that $ι_X$ and $ι_Y$ satisfy particularly strong preservation properties. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of $Δ_1$-completions and appropriate homomorphisms. We formalize the theory of extension polarities and prove a duality principle to the effect that if a statement is true for all extension polarities then so too must be its dual statement.
We use model theoretic techniques to construct explicit first-order axiomatizations for the classes of posets that can be represented as systems of sets, where the order relation is given by inclusion, and existing meets and joins of specified countable cardinalities correspond to intersections and unions respectively.
Given a poset $P$, a join-specification $\mathcal U$ for $P$ is a set of subsets of $P$ whose joins are all defined. The set $\mathcal I_{\mathcal U}$ of downsets closed under joins of sets in $\mathcal U$ forms a complete lattice, and is, in a sense, the free $\mathcal U$-join preserving join-completion of $P$. The main aim of this paper is to address two questions. First, given a join-specification $\mathcal U$, when is $\mathcal I_{\mathcal U}$ a frame? And second, given a poset $P$, what is the structure of its set of frame-generating join-specifications? To answer the first question we provide a number of equivalent conditions, and we use these to investigate the second. In particular, we show that the set of frame-generating join-specifications for $P$ forms a complete lattice ordered by inclusion, and we describe its meet and join operations. We do the same for the set of `maximal' such join-specifications, for a natural definition of `maximal'. We also define functors from these lattices, considered as categories, into a suitably defined category of frame-completions of $P$, and construct right adjoints for them.
Let $m$ and $n$ be cardinals with $3\leq m,n\leqω$. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than $m$ and $n$ respectively cannot be finitely axiomatized.
Given a poset $P$ and a standard closure operator $Γ:\wp(P)\to\wp(P)$ we give a necessary and sufficient condition for the lattice of $Γ$-closed sets of $\wp(P)$ to be a frame in terms of the recursive construction of the $Γ$-closure of sets. We use this condition to show that given a set $\mathcal{U}$ of distinguished joins from $P$, the lattice of $\mathcal{U}$-ideals of $P$ fails to be a frame if and only if it fails to be $σ$-distributive, with $σ$ depending on the cardinalities of sets in $\mathcal{U}$. From this we deduce that if a poset has the property that whenever $a\wedge(b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(ω,3)$-representation. This answers a question from the literature.
A poset is $(ω,C)$-representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of $(ω,C)$-representable posets cannot be axiomatized in first order logic using the standard language of posets. We generalize this result to $(α,β)$-representable posets for certain values of $α$ and $β$.
A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals $α$ and $β$ a poset is said to be $(α,β)$-representable if an embedding into a field of sets exists that preserves meets of sets smaller than $α$ and joins of sets smaller than $β$. We show using an ultraproduct/ultraroot argument that when $2\leqα,β\leq ω$ the class of $(α,β)$-representable posets is elementary, but does not have a finite axiomatization in the case where either $α$ or $β=ω$. We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.
Using the well-known equivalence between meet-completions of posets and standard closure operators we show a general method for constructing meet-completions for isotone poset expansions. With this method we find a meet-completion for ordered domain algebras which simultaneously serves as the base of a representation for such algebras, thereby proving that ordered domain algebras have the finite representation property. We show that many of the equations defining ordered domain algebras are preserved in this completion but associativity, (D2) and (D6) can fail.