Aldo Ursini

Bio: Aldo Ursini is an academic researcher from University of Siena. The author has contributed to research in topics: Variety (universal algebra) & Congruence relation. The author has an hindex of 13, co-authored 23 publications receiving 671 citations. Previous affiliations of Aldo Ursini include Darmstadt University of Applied Sciences & Stellenbosch University.

Ideals in universal algebras

Abstract: In many familiar classes of algebraic structures kernels of congruence relations are uniquely specified by the inverse images q~-l(0)= {x [ q~(x)= 0} of a specified constant 0. On the one hand, q~-l(0) is nothing else but the 0-class of the kernel congruence of q~, on the other hand q~-~(0) can be axiomatized intrinsically, namely q l (0 ) is an ideal (in rings, Boolean algebras, or more generally in Heyt ing algebras), a normal subgroup, resp. normal subloop (in groups, resp. loops) or a filter (in Implicat ion algebras or Boolean algebras again, where 0 is replaced by the unit). In this paper we investigate common features of all the above structures by using a general notion of " ideal" , which makes sense in all universal algebras having a constant 0 and which specializes to the familar concepts of ideal, normal subgroup or filter in each of the algebras quoted above. In all universal algebras the 0-classes of congruence relations are easily seen to be ideals, but we shall require that conversely each ideal is the 0-class of a unique congruence relation. Such algebras, or ra ther classes of algebras with this proper ty will be called "classes with ideal determined congruences" or shortly ideal determined. In Part 1, after presenting the precise definitions, we shall show that the ideal determined varieties are characterized by a Mal 'cev condition, which turns out to be a combination of Fichtner 's condition for 0-regularity together with a ternary te rm r(x, y, z) which is a weakened form of Mal 'cev 's permutabil i ty term. From a result of Hagem ann it follows that ideal-determined varieties have modular congruence lattices, so the theory of commutators becomes readily available. In

On subtractive varieties, I

Abstract: 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, (ie 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

Logic and Algebra

Abstract: "Invited Papers Logic of Proofs with Complexity Operators, S. ArtEmov and A. Chuprina Beyond the s-Semantics: A Theory of Observables, M. Comini and G. Levi The Logic of Commuting Equivalence Relations, D. Finberg, M. Mainetti, and G.-C. Rota Proof-Nets: The Parallel Syntax for Proof-theory, J.-Y. Girard Magari and Others on GOdel's Ontological Proof, P. Hajek Finitely Generated Magari Algebras and Arithmetic, L. Hendriks and D. de Jongh The Butterfly and the Serpent, J. Lambek Adjoints in and Among Bicategories, F. William Lawvere Exponential Algebra, A. Macintyre Categorical Equivalences for Varieties, R. McKenzie Boolean Universal Algebra, A.F. Pixley Restructuring Mathematical Logic: An Approach Based on Peirce's Pragmatism, R. Wille The Development of Research in Algebra in Italy from 1850 to 1940, G. Zappa Contributed Papers A Criterion to Decide the Semantic Match Problem, G. Aguzzi and U. Modigliani Remarks on Magari Algebras of PA and IDelta0+EXP, L. Beklemishev Undecidability in Weak Membership Theories, D. BellE and F. Parlamento Infinite Lambda-Calculus and Non-sensible Models, A. Berarducci A Computer Study of 3-Element Groupoids, J. Bremen and S.N. Burris Ideal Properties of Congruencies, I. Chajda Dualisability in General and Endodualisability in Particular, B.A. Davey Hyperordinals and Nonstandard Alpha-Models, M. Di Nasso Some Notes on Subword Quantification and Induction Thereof, F. Ferreira Research in Automated Deduction as a Basis for a Probabilistic Proof-theory, P. Forcheri, P. Gentilini, and M.T. Molfino Idempotent Simple Algebras, K. Kearnes A Revision of the Mathematical Part of Magari's Paper on ""Introduction to Metamorality"", R. Magari and G. Simi Some Aspects of the Categorical Semantics for the Polymorphic Lambda-Calculus, M.E. Maietti Reflection Using the Derivability Conditions, S. Matthews and A.K. Simpson Stone Bases, Alias the Constructive Content of Stone Representation, S. Negri On k-Permutability for Categories of T-Algebras, M.C. Pedicchio Weak vs. Strong Boethius' Thesis: A Problem in the Analysis of Consequential Implication, C. Pizzi A New and Elementary Method to Represent Every Complete Boolean Algebra, G. Sambin On Finite Intersections of Intermediate Predicate Logics, D. Skvortsov A Completeness Theorem for Formal Topologies, A. Valentini "

Ideals and other generalizations of congruence classes

Abstract: In the general context of ideals in universal algebras, we study varietal properties connected with ideals that are equivalent both to Mal'cev conditions and congruence properties such as 0-regularity, O-permutability, etc.

Ideal determined categories

Abstract: Nous clarifions le role de l'axiome de Hofmann dans la definition "a l'ancienne mode" de categorie semi-abelienne. En enlevant cet axiome nous obtenons la contrepartie categorique de la notion de variete avec determination des ideaux ("ideal determined") d'algebres universelles ― que nous appelons alors categorie ideal determinee. En utilisant des contre-exemples provenant de l'algebre universelle nous pouvons conclure qu'il y a des categories ideal determinees qui ne sont pas des categories de Mal'tsev. Nous montrons aussi qu'il existe des categories de Mal'tsev ideal determinees qui ne sont pas semi- abeliennes.

Commutator theory for congruence modular varieties

Abstract: Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgroups of a group) is the (normal) subgroup generated by all the commutators [x, y] with x ∈ M, y ∈ N. Thus we have a binary operation in the lattice of normal subgroups. This binary operation, in combination with the lattice operations , carries much of the information about how a group is put together. The operation is also interesting in its own right. It is a com-mutative, monotone operation, completely distributive with respect to joins in the lattice. There is an operation naturally defined on the lattice of ideals of a ring, which has these properties. Namely, let [J, K] be the ideal generated by all the products jk and kj, with j ∈ J and k ∈ K. The congruity between these two contexts extends to the following facts: [M, M] is the smallest normal subgroup U of M for which M/U is a commutative group; [J, J] is the smallest ideal K of J for which the ring J/K is a commutative group; that is, a ring with trivial multiplication. Now it develops, amazingly, that a commutator can be defined rather naturally in the congruence lattices of every congruence modular variety. This operation has the same useful properties that the commutator for groups (which is a special case of it) possesses. The resulting theory has many general applications and, we feel, it is quite beautiful. In this book we present the basic theory of commutators in congruence modular varieties and some of its strongest applications. The book by H. P. Gumm [41] offers a quite different approach to the subject. Gumm developed a sustained analogy between commutator theory and affine geometry which allowed him to discover many of the basic facts about the commutator. We take a more algebraic approach, using some of the shortcuts that Taylor and others have discovered. 1 2 INTRODUCTION Historical remarks. The lattice of normal subgroups of a group, with the commutator operation, is a lattice ordered monoid. It is a residuated lattice …

Light linear logic

Abstract: We are seeking a ``logic of polytime'', not yet one more axiomatization, but an intrinsically polytime system. Our methodological bias will be to consider that the expressive power of a system is the complexity of its cut-elimination procedure, and we therefore seek a system with a polytime complexity for cut-elimination (to be precise: besides the size of the proof, there will be an auxiliary parameter, the depth, controlling the degree of the polynomial). This cannot be achieved within classical or intuitionistic logics because of structural rules, especially contraction: this is why the complexity of cut-elimination in all extant logical systems (including the standard version of linear logic which controls structural rules without forbidding them) is catastrophic, elementary (towers of exponentials) or worse. Light Linear Logic (LLL) is a purely logical system with a more careful handling of structural rules: this system is strong enough to represent all polytime functions, but cut-elimination is (locally) polytime. With LLL, our control over the complexity of cut-elimination improves greatly. But this is not the only potentiality of LLL: why not transform it into a system of mathematics and try to formalize ``polytime mathematics'' in the same way as Heyting arithmetic formalizes constructive mathematics? The possibility is clearly open, since LLL admits extensions into a naive set-theory, with full comprehension, still with polytime cut-elimination. This system admits full induction on data types, which shows that, within LLL, induction is compatible with low complexity.

Hybrid clustering analysis using improved krill herd algorithm

Abstract: In this paper, a novel text clustering method, improved krill herd algorithm with a hybrid function, called MMKHA, is proposed as an efficient clustering way to obtain promising and precise results in this domain. Krill herd is a new swarm-based optimization algorithm that imitates the behavior of a group of live krill. The potential of this algorithm is high because it performs better than other optimization methods; it balances the process of exploration and exploitation by complementing the strength of local nearby searching and global wide-range searching. Text clustering is the process of grouping significant amounts of text documents into coherent clusters in which documents in the same cluster are relevant. For the purpose of the experiments, six versions are thoroughly investigated to determine the best version for solving the text clustering. Eight benchmark text datasets are used for the evaluation process available at the Laboratory of Computational Intelligence (LABIC). Seven evaluation measures are utilized to validate the proposed algorithms, namely, ASDC, accuracy, precision, recall, F-measure, purity, and entropy. The proposed algorithms are compared with the other successful algorithms published in the literature. The results proved that the proposed improved krill herd algorithm with hybrid function achieved almost all the best results for all datasets in comparison with the other comparative algorithms.

The structure of residuated lattices

Abstract: A residuated lattice is an ordered algebraic structure such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

A Survey of Residuated Lattices

Abstract: Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.