Welcome to Paolo Lipparini's home page. Willkommen. Bienvenues. Benvenuti. Witajcie. ¡Bienvenidos! Bem-vindos. Benibenius. Dobró požálovat’. Yo-koso. Isten hozott.

e-mail: [the first 8 letters of my family name] (a t) axp (d ot) mat ( d ot) uniroma2 (d ot) it
Alternative (emergency) e-mail: lpppla01 (at ) uniroma2 (dot) it

• informazioni per gli STUDENTI (only italian)

My main mathematical interests are Ultrafilters, General Algebra, General Topology, Model Theory, Set Theory, Ordered Sets and Foundations of Mathematics.

Here you can download some manuscripts, a very short curriculum (italiano, english) , some papers (not up-to-date) slides from some talks, as well as some older stuff.
For all of my works, an important disclaimer is in charge, regarding possible misuses of the reference lists.

You can also find my contact informations, and some (very disparate) links.

Manuscripts ( (many) more manuscripts are available at arXiv.org)

• Initial $\lambda$-compactness in linearly ordered spaces
(Topology: compactness of products of linearly ordered spaces; ultrafilter convergence; variations on pseudocompactness)
For linearly ordered topological spaces, the notions of (weak) initial lambda compactness and of lambda boundedness (and hence all intermediate notions) coincide.
As a consequence, every product of initially lambda compact linearly ordered spaces is still initially lambda compact.

• Compactness of powers of omega
(Topology: compactness of products; Ultrafilters decomposable; Model Theory: kinds of nonstandard elements, compactness of infinitary languages)
We characterize exactly the compactness properties of the product of kappa copies of the space of the natural numbers with the discrete topology. Some results extend to products of possibly uncountable regular cardinals with the order topology.

• Ordinal compactness
(Topology, covering properties; ordinal arithmetic)
We introduce new compactness notions, defined in terms of order types of sequences, rather than cardinalities. Such ordinal compactness properties turn out to behave in a much more varied way, in comparison with classical cardinal compactness.
We deal particularly with spaces of small cardinality, with T1 spaces, and with disjoint unions.

Recent papers (not up to date)
(An updated list can be found using MathSciNet, zbMATH etc.; Journal-refs on arXiv.org are updated, too)
(Notice: there might be differences between the version available here and the published version. When dealing with important matters, please always refer to the published version.)

• Representable tolerances in varieties
(Universal Algebra, general varieties)
We discuss two possible ways of representing tolerances: as a homomorphic image of some congruence, and as the relational composition of some compatible relation with its converse. The relationship between these two representations are discussed.
A very simple proof is presented, showing that any tolerance on some lattice L is the image of some congruence on a subalgebra of L × L
Acta Sci. Math. (Szeged), Volume 79, Numbers 1-2, 2013.

• Tolerances as images of congruences in varieties defined by linear identities
(with Ivan Chajda, Gábor Czédli, Radomir Halas)
(Universal Algebra, general varieties)
Algebra Universalis published online

• A very general covering property
(Topology, compact-like and pseudocompact-like covering properties, accumulation properties, convergence)
Covering notions are exactly equivalent to accumulation notions.
Commentationes Mathematicae Universitatis Carolinae, vol. 53 (2012), issue 2, pp. 281-306

• Every weakly initially m-compact topological space is mpcap
(Topology, compact-like and pseudocompact-like covering properties)
The statement in the title solves a problem by T. Retta. Incidentally, some more general results are obtained.
Czechoslovak Mathematical Journal, vol. 61, no. 3 (2011), pp. 781-784.

• More generalizations of pseudocompactness
(Topology, covering properties, Ultrafilters)
We introduce a covering notion which encompasses many generalizations of pseudocompactness introduced before by many authors, including Comfort, Frolik, Ginsburg, Negrepontis and Saks.
Ultrafilters are applied to the study of these notions, especially in connection with products.
Topology and its Applications Volume 158 (2011), 1655-1666.

• Some compactness properties related to pseudocompactness and ultrafilter convergence
(Topology, covering properties, convergence)
We relativize many notions of compactness and convergence to families of subsets. Some results on pseudocompactness and D-pseudocompactness seem to be new.
Topology Proceedings, Volume 40 (2012), 29-51.

• More on regular and decomposable ultrafilters in ZFC
(Ultrafilters, Set Theory, a few applications to Model Theory and Topology)
A manuscript on regularity properties of ultrafilters. Also dealing with decomposability and other stuff. It tries to survey all known results in the field.
Mathematical Logic Quarterly, Volume 56, vol. 4, pagg. 340-–374 2010.

• Decomposable ultrafilters and possible cofinalities (PDF) (DVI) (PS)
(Ultrafilters, Set Theory, applications to Model Theory and Topology)
We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. For example, we show that an ultrafilter is $(\lambda,\lambda)$-regular if and only if it is either $\lambda^+$-decomposable or $cf (\lambda)$-decomposable. We give applications to $[\lambda, \lambda]$-compactness of topological spaces and of abstract logics.
Notre Dame Journal of Formal Logic 49, 307--312 (2008).

• Terms which are Mal'cev modulo some functions (PDF)
(Universal Algebra, no assumption on the variety)
We derive consequences from the existence of a term which satisfies Mal'cev identities (characterizing permutability) modulo two functions $F$ and $G$ from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal'cev term modulo $F$ and $G$.
Algebra Universalis 58 (2008) 249-262.

• Tolerance intersection properties and subalgebras of squares (PDF) (DVI) (PS)
(Universal Algebra, no assumption on the variety)
We prove that if every subalgebra of A^2 is congruence modular then A satisfies the tolerance identity wTIP:
$\Gamma^*\cap\Theta^*=(\Gamma^*\cap \Theta)^*$.
wTIP is equivalent to Gumm's Shifting Principle.
Published in: Andretta, Alessandro; Kearnes, Keith; Zambella, Domenico (editors), Logic colloquium 2004. Cambridge University Press; Association for Symbolic Logic. Lecture Notes in Logic 29, 109-122 (2008).

• Every m-permutable variety satisfies the congruence identity $\alpha \beta_h= \alpha \gamma_h$ (PDF)
(Universal Algebra, m-permutable varieties )
Proc. Amer. Math. Soc. 136 (2008), no. 4, 1137--1144.

• From congruence identities to tolerance identities (PDF) (DVI) (PS)
(Universal Algebra, general varieties)
We show that, under certain conditions, a variety satisfies a given congruence identity if and only if it satisfies the same tolerance identity. We try to clarify the role of graphs in the study of Mal'cev conditions.
Acta Sci. Math. (Szeged) 73 (2007), 31-51

• Compact factors in finally compact products of topological spaces (PDF)
(Topology, Set Theory)
We show that if a product of topological spaces satisfies some compactness property then the factors satisfy a stronger compactness property, except for a small number of factors.
Topology and its Applications Vol 153 (9) 1365-1382 (2006).

• If every subalgebra of A^4 is congruence modular then A satisfies the tolerance identity
$\Gamma^*\cap\Theta^*=(\Gamma\cap(\Theta\circ \Theta))^*$
(PDF)
(DVI) (PS)
(Universal Algebra, no assumption on the variety)
Shortened and published as: A local proof for a tolerance intersection property, Algebra Universalis Vol. 54 (3) 273-277 (2005) - external link to the published version.

• Optimal Mal'tsev conditions for congruence modular varieties (PDF) (DVI) (PS)
(joint work with G. Czedli and E. Horvath ).
(Universal Algebra, modular varieties)
There we get Optimal Mal'tsev conditions for congruence modular varieties, we show that such varieties satisfy higher Arguesian identities and we get a very strong version of Gumm's "congruence modularity is distributivity composed with permutability".
External link to the published version, Algebra Universalis 53 (2-3) 267-279 (2005).

Older manuscripts and papers.

- Scanned from the originals (I plan to add more)

• The compactness spectrum of abstract logics, large cardinals and combinatorial principles
(Model Theory, Set Theory, Ultrafilters)
Bollettino U.M.I. (7) 4-B, 875-903 (1990).

• Difference terms and commutators
(Universal Algebra, no assumption on the variety)
Unpublished, but occasionally quoted.
K. Kearnes and A. Szendrei have obtained strong related results. See The relationship between two commutators .

• Congruence identities satisfied in n-permutable varieties
(Universal Algebra, m-permutable varieties)
Bollettino U.M.I. (7) 8-B, 851-868 (1994).

• About some generalizations of ($\lambda$, $\mu$)-compactness
(Model Theory, Set Theory, Ultrafilters)
Proceedings of the 5th Easter Conference on Model Theory (Wendisch Rietz, 1987), Seminarber., Humboldt-Univ. Berlin, Sekt. Math. 93, 139--141 (1987).
Reviewed here by Zentralblatt MATH.

- Other manuscripts.

• A new proof that n-permutable varieties satisfy lattice identities (DVI)
(Universal Algebra, m-permutable varieties)
This proof does not use commutator theory.

• Addendum to "Existentially complete closure algebras"
(Algebra, Topology)

- In italiano.

• Logica Matematica e valore della scienza

• Una breve introduzione alla teoria del commutatore in algebra generale

• Problemi matematici risolti con metodi logici

• ETICA DEGLI AUTOVALORI

• Il Principio dei Calzini Spaiati

Slides

• Slides and abstract of the talk at Ultramath2008 , Pisa, about decomposability of ultrafilters and applications to Topology, Set Theory and Model Theory.

• Slides of the talk at XXIV Incontro di Logica , (Bologna february 2011) about "ordinal compactness" for topological spaces. More details on arXiv:1012.4737 .

• Slides and abstract of the talk at XXV Incontro di Logica (AILA meeting) , Pisa, 14-17 aprile 2014, about "Weak and local versions of measurability" (some inaccuracies might be present). More details on arXiv:1404.1586 .

• Music
• La via matematica all'umorismo (una raccolta di barzellette matematiche molto accurata. Altri link qui)
• ROARS sito sui problemi dell'’università e della ricerca

How to contact me

You can write me at the following addresses:
[the first 8 letters of my second name] (a t) axp (d ot) mat ( d ot) uniroma2 (d ot) it

Dipartimento di Matematica
Universita' di Tor Vergata
Viale della Ricerca Scientifica
I-00133 ROMA

phone (office) 39-6-72594847

Arrivederci. See you. Auf Wiedersehen. Au revoir. Sayonara. Aloha. Hasta luego. Até logo. A si biri.