Aldo Ursini

On subtractive varieties, I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a “classical” ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.

On subtractive varieties II: General properties

As a sequel to [23] we investigate ideal properties focusing on subtractive varieties. After having listed a few basic results, we give several characterizations of the commutator of ideals and prove, for example, that it commutes with finite direct products. We deal with the ideal extension property (IEP) and with related commutator properties, showing for instance that IEP implies that the commutator commutes with restriction to subalgebras. Then we characterize varieties with distributive ideal lattices and relate this property with (a form of) equationally definable principal ideals and with IEP. Then, at the other extreme, we deal with Abelian and Hamiltonian properties (of ideals and congruences), giving for example a purely ideal theoretic characterization of varieties of Abelian groups with linear operations. To finish with, we present a few examples aiming at vindicating our work.

Ideals and Clots in Pointed Regular Categories

We clarify the relationship between ideals, clots, and normal subobjects in a pointed regular category with finite coproducts.

Franco Montagna (1948---2015)

Two powerful streams in the first half of last century impeded the development of Mathematical Logic (ML) in Italy after Peano and his school. From one side, Italian culture was dominated by the local version of Idealistic Philosophy1 (Gentile was Minister for Education under Mussolini) with its emphasis on the superiority of philosophy over science. From the other side, the reaction of mathematicians, in particular, was to establish their point by stressing the applicability of Mathematics (Vito Volterra was the head of the National Research Council in the same period). ML was smashed in between the two damnations: “Mathematica sunt, non leguntur–Philosophica sunt, non leguntur”. For instance, the influential mathematician and philosopher Federigo Enriques (1871–1946)2 expressed dramatic judgments about the developments of ML and of Set Theory well into the thirties. For him, while at its origins in the hands of Bolzano, Dedekind or Cantor the mathematical investigation of the Foundations was a decisive step for Mathematics, the abstraction in the current ML of his time was deemed as “transcendent” or “metaphysical”; for instance, Enriques never accepted the axiom of choice or any unbridled usage of actual infinity; he apparently did not take seriously Symbolic Logic: on that he possibly agreed with Benedetto Croce, in considering it as a child’s game. There was no teaching of ML courses in the Universities and preciously few papers

Weighted commutators in semi-abelian categories

Abstract We introduce new notions of “weighted centrality” and “weighted commutators” corresponding to each other in the same way as centrality of congruences and commutators do in the Smith commutator theory. Both the Huq commutator of subobjects and Pedicchioʼs categorical generalization of Smith commutator are special cases of our weighted commutators; in fact we obtain them by taking the smallest and the largest weight respectively. At the end of the paper we briefly consider the universal-algebraic context in connection with an older work of the third author on the ideal theory version of the commutator theory.

A modal calculus analogous to K4W, based on intuitionistic propositional logic, Iℴ

This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the “provability” predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way.Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.

Normal Subalgebras, I

We extend the notions of normal subalgebras, clots and ideals of an algebra A in a variety of (universal) algebras, from the familiar case of a single constant to the case of any number of constants. The first idea is that a subalgebra of A is normal when it is the inverse image under some morphism of the subalgebra generated by constants in the target. We argue that a better approach is obtained by considering pullbacks of γB and g : A → B, where g : A → B is some morphism and γB is the morphism from the initial algebra of the variety to B. Examples are shown in Heyting algebras, boolean algebras and unitary rings. Ideals and clots are generalizations of this notion, defined instead by closure under derived operations which have the right behavior on constants. There are several characterizations of these notions; some of them aiming at a categorical generalization. We deal with an (extended) notion of subtractivity, showing that it implies that ideals coincide with normal subalgebras, and it is connected with notions of coherence of congruences, allowing a characterization of protomodular varieties.

On the set of ‘Meaningful’ sentences of arithmetic

SummaryI give several characterizations of the setV0 proposed in [3] as the set of meaningful and true sentences of first order arthimetic, and show that in Peano arithmetic theΣ2 completeness ofV0 is provable.

Cosets in universal algebra

The aim of this paper is to show that a close investigation of congruence classes of an algebra may give useful information on varieties to which the algebra happens to belong. In principle our approach is quite distinct from that of Wille [9] since we are interested in giving a “natural” structure to the set of congruence classes containing a given element of the algebra, and to relate the properties of this structure to general features of the variety generated by the algebra. It comes out that such kind of results is better understood if we consider the set of congruence classes containing a given element as a subset of a generally richer family of subset of the algebra--we will call these “cosets” of the algebra-which in fact is endowed with the very natural structure of an algebraic lattice. It is easy to see that in many classical cases cosets are very-well-known animals: in the case of groups we get cosets (with respect to normal subgroups), in rings we get ideals (when the fixed element is zero), etc. One main result sounds as follows: for a variety V the fact that, for each algebra JZZ E V, for each LZE A the natural map a/ which sends a congruence R E Con(,&‘) into a/R is a lattice homomorphism is equivalent to congruence permutability. Our notation is, more or less, standard; for general background in Universal Algebra we refer to [ 1, 41.

MORE IDEALS IN UNIVERSAL ALGEBRAS

Publisher Summary This chapter describes some ideals in universal algebras. If K is a variety of universal algebras having a nullary operation or equationally defined constant 0, then the notion of a 0-ideal for algebras in K can be generalized. The chapter presents some results concerning this concept mainly for the case when K is ideal-determined, which means that every 0-ideal is the class of 0 for exactly one congruence. It describes the generalization of this notion to get an f -ideal of A , where K is a variety, and A∈K and f is an arbitrary fundamental operation or a polynomial on A . The chapter further presents the notion of an f -normal subset of A n+1 , where f is n -ary, and also explores some of their properties and connections between them.