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Paolo Lipparini

Publications and source records attributed to Paolo Lipparini.

At least 19 recordsLinked to original sources
arXiv2026

A hyperamalgamation property

The superamalgamation property is a strengthening of the amalgamation property and has found many applications in algebraic logic, more recently, also in model theory. Here we introduce an even stronger notion we call the hyperamalgamation property. We show that lattices, semilattices, Boolean algebras and dual Heyting algebras have the hyperamalgamation property. As a first application, we obtain the amalgamation property for semilattices, Boolean algebras and Heyting algebras with further operators.

math.LO↗
arXiv2026

Noncommutative infinitary semigroups

Various kinds of infinitary operations satisfying forms of associativity have been considered in the literature by various authors, including A. Tarski, C. Karp, J. H. Conway, D. Krob, N. Bedon, and C. Rispal. Applications include the arithmetics of binary relations, quasigroups, and automata theory, We present a general definition for an infinitary noncommutative partial semigroup; the definition extends and encompasses all the previous notions. In particular, we show that new phenomena occur in the noncommutative case, giving rise to a somewhat richer (and, by the way, more difficult) theory.

math.GR↗
arXiv2026

One-sided inverses in noncommutative infinitary semigroups

In a former paper we introduced partial infinitary noncommutative semigroups and showed, among other, that significant differences arise in comparison with the commutative case, previously studied in the literature. For example, in the commutative case we cannot have an infinitary identity $e$ together with two elements $a \not= e$, $b \not= e$ such that $ab= e$, just under the assumption that the countable product $abababa\dots$ is defined. Here we show that this is possible in the noncommutative case, actually, we can have an infinitary semigroup on a countable set with a complete identity and such that the operation is defined for every indexed linearly ordered set.

math.GR↗
arXiv2026

Ordinal semigroups

In a previous paper we introduced a version of associativity for a partial infinitary operation. We prove here that if $\gamma$ is an infinite ordinal and some associative infinitary operation is defined for all sequences indexed by ordinals $ \leq \gamma$, then such an operation can be uniquely expanded to apply to every sequence indexed by any ordinal of cardinality $ |\gamma |$. In particular, if some associative operation is defined for all finite sequences as well as for all $ \omega$-indexed sequences, then the operation can be uniquely expanded to apply to every sequence indexed by a countable ordinal.

math.LO↗
arXiv2026

A Ring structure on the Class of Combinatorial Games

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games form a Group. However, Conway product is not well defined on equivalence classes of arbitrary games (though it is well defined for surreals). We consider an equivalence relation finer than Conway's and show that under such a relation combinatorial games actually form a Ring. We hint to other possible relations on the Class of combinatorial games.

math.CO↗
arXiv2026

Monotone infinitary operations on ordinals (extended version)

We define and study an $ ω$-ary operation on the class of the ordinals, which is strictly monotone in many significant cases (by an elementary argument, there is no fully strictly monotone infinitary operation on ordinals). We compare the operation with the finitary Hessenberg natural sum, which is the smallest finitary strictly monotone operation on each argument. We also compare it with other infinitary generalizations of Hessenberg sum. We provide order-theoretical characterizations of our operation, both as the rank of sequences in an appropriate well-founded order, and as a mixed (or shuffled) sum of the ordinals in the sequence. The latter means that such an infinitary sum is the largest realization as an order-preserving disjoint union of copies of the summands, under some boundedness restriction. The former characterization can be recast in terms of combinatorial games, leading to the problem whether the operation can be extended to the class of Conway surreal numbers.

math.LO↗
arXiv2026

Hypercontact semilattices

Contact Boolean algebras are one of the main algebraic tools in region-based theory of space. T. Ivanova provided strong motivations for the study of merely semilattices with a contact relation. Another significant motivation for considering an even weaker underlying structure comes from event structures with binary conflict in the theory of concurrent systems in computer science. All the above-hinted notions deal with a binary contact relation. Several authors suggested the more general study of $n$-ary ``hypercontact'' relations and noticed that, in general, a hypercontact relation cannot be retrieved from just a binary contact relation. A similar evolution occurred in the study of the just mentioned event structures in computer science. In an effort to unify the above lines of research, in this paper we study join semilattices with a hypercontact relation. We provide representation theorems into Boolean algebras, with or without overlap hypercontact relation. With a single exception, our proofs are choice-free. We also present several examples and problems; in particular, we briefly discuss some connections with event structures and hypergraphs.

math.LO↗
arXiv2025

A model theory of topology

An algebraization of the notion of topology has been proposed more than seventy years ago in a classical paper by McKinsey and Tarski. However, in McKinsey and Tarski's setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation $ \sqsubseteq $ defined by $a \sqsubseteq b$ if $a$ is contained in the topological closure of $b$. A specialization poset is a partially ordered set endowed with a further coarser preorder relation $ \sqsubseteq $. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate settings, even far removed from topology.

math.GN↗
arXiv2025

Semilattices with a congruence

A specialization semilattice is a semilattice together with a coarser preorder satisfying a compatibility condition. We show that the category of specialization semilattices is isomorphic to the category of semilattices with a congruence, hence equivalent to the category of semilattice epimorphisms. Guided by the above example, we recall an ``internal'' characterization of surjective homomorphisms between general relational systems.

math.RA↗
arXiv2025

Infinite sums of combinatorial games (Dadaist games)

We propose an interpretation of the infinite sum of combinatorial games. In such an interpretation, plays involve infinite runs, but without loops. The notion of a run is quite natural, but different possibilities arises for the notion of an alternating run.

math.CO↗
arXiv2024

A short way to directed Jónsson terms

We show that a variety with Jónsson terms $t_1, \dots, t_{n-1}$ has directed Jónsson terms $d_1, \dots, d_{n-1}$, for the same value of the indices, solving a problem raised by Kazda et al.. Refined results are obtained for locally finite varieties.

math.RA↗
arXiv2024

Series of combinatorial games

We present a definition for the sum of a sequence of combinatorial games. This sum coincides with the classical sum in the case of a converging sequence of real numbers and with the infinitary natural sum in the case of a sequence of ordinal numbers. We briefly discuss other possibilities, such as the string limit, some "magical" variants of Hackenbush, as well as "Dadaist" infinite sums, which allow transfinite runs, while still being loopfree.

math.CO↗
arXiv2024

Finitely Based Congruence Varieties

We show that for a large class of varieties of algebras, the equational theory of the congruence lattices of the members is not finitely based.

math.RA↗
arXiv2023

Ivanova contact join-semilattices are not finitely axiomatizable

We show that the class of Contact join-semilattices, as introduced by T. Ivanova, is not finitely axiomatizable. On the other hand, a simple finite axiomatization exists for the class of those join semilattices with a weak contact relation which can be embedded into the reduct of a Boolean algebra (equivalently, a distributive lattice) with a weak contact relation.

math.LO↗
arXiv2023

Pairs of partial orders and the amalgamation property

We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fra\"ıssé limits and, in many cases, an $ ω$-categorical model completion with quantifier elimination. The same applies to Kronheimer and Penrose's causal spaces. On the other hand, Urquhart doubly ordered sets do not have the amalgamation property. Our main tool is the superamalgamation property (the strong amalgamation property is not enough), thus we provide further arguments suggesting the usefulness of the superamalgamation property also in pure model theory, not only in algebraic logic.

math.LO↗
arXiv2023

Contact posets

We study contact posets and show that every contact poset can be embedded into a Boolean poset with overlap contact relation. Contact posets and (nonadditive) contact semilattices have the superamalgamation property, Fra\"ıssé limits and model completion. Some results apply to event structures with binary conflict, as introduced in computer science.

math.LO↗
arXiv2023

Contact semilattices

We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation. A similar characterization is proved with respect to Boolean algebras and distributive lattices with weak contact, not necessarily additive, nor overlap.

math.LO↗