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Paolo Lipparini

Publications and source records attributed to Paolo Lipparini.

87 records · Page 5Linked to original sources
arXiv2008

More Results on Regular Ultrafilters in ZFC

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.

math.LO↗
arXiv2008

Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. IV

We extend to singular cardinals the model-theoretical relation $λ\stackrelκ{\Rightarrow} μ$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics.

math.LO↗
arXiv2007

Mal'cev conditions from graphs for congruences and tolerances

In a previous paper (From congruence identities to tolerance identities, in print on Acta Sci. Math. Szeged) we showed that, under certain conditions, a variety satisfies a given congruence identity if and only if it satisfies the same tolerance identity. Moreover, we mentioned that a parallel result holds for Mal'cev conditions arising from graphs. In the present note we give detailed definitions and state these more general results.

math.RA↗
arXiv2006

Towards Commutator theory for relations. IV

We provide more characterizations of varieties having a term Mal'cev modulo two functions $F$ and $G$. We characterize varieties neutral in the sense of $F$, that is varieties satisfying $R \subseteq F(R)$. We present examples of global operators satisfying the homomorphism property, in particular we show that many known commutators satisfy the homomorphism property.

math.RA↗
arXiv2006

A connection between decomposable ultrafilters and possible cofinalities. II

We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ λ$ is a singular cardinal, $ λ' < λ$, and the ultrafilter $D$ is $ κ$-decomposable for all regular cardinals $ κ$ with $ λ' < κ< λ$. Then $D$ is either $ λ$-decomposable, or $ λ^+$-decomposable. We give applications to topological spaces and to abstract logics.

math.LO↗
arXiv2006

A connection between decomposability of ultrafilters and possible cofinalities

We introduce the decomposability spectrum $K_D=\{λ\geq ω| D \text{is} λ\text{-decomposable}\}$ of an ultrafilter $D$, and show that Shelah's $\pcf$ theory influences the possible values $K_D$ can take. For example, we show that if $\aaa$ is a set of regular cardinals, $μ\in \pcfa$, the ultrafilter $D$ is $|\aaa |^+$-complete and $K_D \subseteq \aaa$, then $μ\in K_D$. As a consequence, we show that if $ λ$ is singular and for some $ λ' < λ$ $K_D$ contains all regular cardinals in $ [λ', λ)$ then: (a) if $\cf λ= ω$ then either $ λ\in K_D$, or $ λ^+ \in K_D$; and (b) if $D$ is $(\cf λ)^+$-complete then $ λ^+ \in K_D$, and $\pp (λ)= λ^+$.

math.LO↗
arXiv2005

Towards commutator theory for relations. III

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.

math.GM↗