arXiv subjects
Publications and source records attributed to Paolo Lipparini.
For every $n$, we evaluate the smallest $k$ such that the congruence inclusion $α(β\circ_n γ) \subseteq αβ\circ_{k} αγ$ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.
Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and Jónsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without Jónsson terms. We prove the quite unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by $1$. It follows that every $r$-modular congruence distributive variety is $r^2-r+2$-distributive.
We study the validity of congruence inclusions of the form $ α( β\circ αγ\circ β\circ \dotsc \circ αγ\circ β) \subseteq αβ\circ αγ\circ αβ\circ \dots$ in congruence modular varieties, with an appropriate number of terms on each side and where juxtaposition denotes intersection. Two different methods using Day and Gumm terms are merged in order to obtain the so far best bounds. We introduce and study other related identities, possibly involving tolerances and admissible relations. We also slightly improve a result by A. Day, to the effect that if $n$ is even, then every variety with $n+2$ Jónsson terms has $2n+1$ Day terms.
We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $\mathcal {V}$ is congruence distributive if and only if the congruence identity $α\cap (β\circ γ\circ β) \subseteq αβ\circ γ\circ αβ\circ γ\dots $ ($k$ factors) holds in $\mathcal {V}$, for some natural number $k$.
Let $α$, $β$, $γ, \dots$ $Θ$, $Ψ, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identity $α(β\circ Θ) \subseteq αβ\circ αΘ\circ αβ$ holds in a variety $\mathcal {V}$, then $\mathcal {V}$ has a majority term, equivalently, $\mathcal {V}$ satisfies $ α(β\circ γ) \subseteq αβ\circ αγ$. The result is unexpected, since in the displayed identity we have one more factor on the right and, moreover, if we let $Θ$ be a congruence, we get a condition equivalent to $3$-distributivity, which is well-known to be strictly weaker than the existence of a majority term. The above result is optimal in many senses, for example, we show that slight variations on the displayed identity, such as $ R (S \circ γ) \subseteq R S \circ R γ\circ R S$ or $R(S \circ T) \subseteq R S \circ RT \circ RT \circ RS$ hold in every $3$-distributive variety. Similar identities are valid even in varieties with $2$ Gumm terms, with no distributivity assumption. We also discuss relation identities in $n$-permutable varieties and present a few remarks about implication algebras.
Let surreal numbers be defined by means of sign sequences. We give a proof that if $S < T$ are sets of surreals, then there is some surreal $w$ such that $S < w < T$. The classical proof is simplified by observing that, for every set $S$ of surreals, there exists a surreal $s$ such that, for every surreal $w$, we have $S<w$ if and only if the restriction of $w$ to the length of $s$ is $ \geq s$. Hence $S < w < T$ if and only if $w$ satisfies the above condition, as well as its symmetrical version with respect to $T$. It is now enough to check that if $S < T$, then the two conditions are compatible.
We show that a variety $\mathcal V$ is congruence distributive if and only if there is some $h$ such that the inclusion (1) $Θ\cap ( σ\circ σ) \subseteq ( Θ\cap σ) \circ ( Θ\cap σ) \circ \dots $ ($h$ factors) holds in every algebra in $\mathcal V$, for every tolerance $Θ$ and every U-admissible relation $σ$. By a U-admissible relation we mean a binary relation which is the set-theoretical union of a set of reflexive and admissible relations. For any fixed $h$, a Maltsev-type characterization is given for the inclusion (1). It is an open problem whether (1) is still equivalent to congruence distributivity when $Θ$ is assumed to be a $U$-admissible relation, rather than a tolerance. In both cases many equivalent formulations for (1) are presented. The results suggest that it might be interesting to study the structure of the set of U-admissible relations on an algebra, as well as identities dealing with such relations.
We introduce the notion of a nest-representable tolerance and show that some results from our former paper "From congruence identities to tolerance identities" [CT] can be extended to this more general setting.
We study an infinite countable iteration of the natural product between ordinals. We present an "effective" way to compute this countable natural product, in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product, this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.
Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $ J _{ \mathcal V} (m) $ be the smallest natural number $k$ such that the congruence identity $α( β\circ γ\circ β\dots ) \subseteq αβ\circ αγ\circ αβ\circ \dots $ holds in $\mathcal V$, with $m$ occurrences of $ \circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $ J _{ \mathcal V} (m) =k$, then $ J _{ \mathcal V} (m \ell ) \leq k \ell $, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $ J _{ \mathcal V} (1)=2 $, that is, $\mathcal V$ is $3$-distributive, then $ J _{ \mathcal V} (m) \leq m $, for every $m \geq 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $ J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are stated at various places.
We present some identities dealing with reflexive and admissible relations and which, through a variety, are equivalent to congruence modularity.
We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages and show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a "mixed sum" (in an order-theoretical sense) of the ordinals in the sequence, in fact, it is the largest mixed sum which satisfies a finiteness condition, relative to the ordering of the sequence. We introduce other infinite natural sums which are invariant under permutations and show that they all coincide in the countable case. Finally, in the last section we use the above infinitary natural sums in order to provide a definition of size for a well-founded tree, together with an order-theoretical characterization in the countable case. The proof of this order-theoretical characterization is mostly independent from the rest of this paper.
We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include local sequential compactness, local Lindelöfness, the local Menger property.
The class $\mathfrak C $ relative to countably compact topological spaces and the class $\mathfrak P$ relative to pseudocompact spaces introduced by Z. Frolík are naturally generalized relative to every topological property. We provide a characterization of such generalized Frolík classes in the broad case of properties defined in terms of filter convergence. If a class of spaces can be defined in terms of filter convergence, then the same is true for its Frolík class.
As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite version of the natural sum has been used in a recent paper by Väänänen and Wang, with applications to infinitary logics. We provide an order theoretical characterization of this operation. We show that this countable natural sum differs from the more usual infinite ordinal sum only for an initial finite "head" and agrees on the remaining infinite "tail". We show how to evaluate the countable natural sum just by computing a finite natural sum. Various kinds of infinite mixed sums of ordinals are discussed.
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $λ$-compact generalized ordered topological spaces is still initially $λ$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.
Local versions of measurability have been around for a long time. Roughly, one splits the notion of $μ$-completeness into pieces, and asks for a uniform ultrafilter over $μ$ satisfying just some piece of $μ$-completeness. Analogue local versions of weak compactness are harder to come by, since weak compactness cannot be defined by using a single ultrafilter. We deal with the problem by restricting just to a subset $P$ of all the partitions of $μ$ into $<μ$ classes and asking for some ultrafilter $D$ over $μ$ such that no partition in $P$ disproves the $μ$-completeness of $D$. By making $P$ vary in appropriate classes, one gets both measurability and weak compactness, as well as possible intermediate notions of "weak measurability". We systematize the above procedures and combine them to obtain variants of measurability which are at the same time weaker and local. Of particular interest is the fact that the notions thus obtained admit equivalent formulations through topological, model theoretical, combinatorial and Boolean algebraic conditions. We also hint a connection with Kat{ě}tov order on filters.
We show that a linearly ordered topological space is initially λ-compact if and only if it is λ-bounded, that is, every set of cardinality $\leq λ$ has compact closure. As a consequence, every product of initially λ-compact linearly ordered topological spaces is initially λ-compact.