Ordinal Sum Research Articles

Analytic Calculi for Logics of Ordinal Multiples of Standard t-Norms

For two propositional fuzzy logics, we present analytic proof calculi, based on relational hypersequents. The logic considered first, called MŁ, is based on the finite ordinal sums of Łukasiewicz t-norms. In addition to the usual connectives—the conjunction ⊙, the implication → and the constant 0—we use a further unary connective interpreted by the function associating with each truth value a the greatest ⊙-idempotent below a. MŁ is a conservative extension of Basic Logic. The second logic, called MΠ, is based on the finite ordinal sums of the product t-norm on (0, 1]. Our connectives are in this case just the conjunction and the implication.

Semi-divisible triangular norms

Semi-divisibility of left-continuous triangular norms is a weakening of the divisibility (i.e., continuity) axiom for t-norms. In this contribution we focus on the class of semi-divisible t-norms and show the following properties: Each semi-divisible t-norm with Ran(n T ) = [0, 1] is nilpotent. Semi-divisibility of an ordinal sum t-norm is determined by the corresponding property of its first component (which can be a proper t-subnorm, too). Finally, negations with finite range derived from semi-divisible t-norms are studied.

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k , and denote by D b ( X ) the bounded derived category of sheaves of finite dimensional k -vector spaces over X . Two posets X and Y are said to be derived equivalent if D b ( X ) and D b ( Y ) are equivalent as triangulated categories. We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z -congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence. For any closed subset Y ⊆ X , we construct a strongly exceptional collection in D b ( X ) and use it to show an equivalence D b ( X ) ≃ D b ( A ) for a finite dimensional algebra A (depending on Y ). We give conditions on X and Y under which A becomes an incidence algebra of a poset. We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite S op . This construction produces many new derived equivalences of posets and generalizes other well-known ones. As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.

Further properties of reversible triangular norms

In this paper we present some properties of discontinuous reversible triangular norms. Two of them seem to be important: (1) a reversible t-norm T which possesses only trivial idempotent elements must be Archimedean. (2) If T is a discontinuous reversible t-norm and has non-trivial idempotent elements, then T is an ordinal sum of two t-noms, the one on the lower part of the unit square dominating Lukasiewcz t-norm, the other being reversible and Archimedean. Consequently, no left-continuous t-norm that is not continuous is reversible. Therefore, the question of reversibility of discontinuous t-norms is reducible to the case when the t-norms in question possess trivial idempotent elements only. Moreover, we have completely characterized the reversibility of T * (the reverse of a reversible t-norm T).

Aglianò–Montagna type decomposition of linear pseudo hoops and its applications

We decompose every linear pseudo hoop as an Aglianò–Montagna type of ordinal sum of linear Wajsberg pseudo hoops which are either negative cones of linear ℓ -groups or intervals in linear unital ℓ -groups with strong unit. We apply the decomposition to present a new proof that every linear pseudo BL-algebra and consequently every representable pseudo BL-algebra is good. Moreover, we show that every maximal filter and every value of a linear pseudo hoop is normal, and every σ -complete linear pseudo hoop is commutative.

Structure of n-uninorms

In this paper we study binary operators on [ 0 , 1 ] which are associative, monotone non-decreasing in both variables and commutative (AMC) with neutral element. In this work, we generalize the concept of neutral element and this generalization gives rise to a new class of AMC binary operators on [ 0 , 1 ] called n- uninorms. n-Uninorms are denoted as U n , where n comes from the generalization of the neutral element. We study the structure of n-uninorms. The structure resembles an ordinal sum structure made up of n uninorms. We characterize some special cases of them based on some continuity considerations and show that t-norms, t-conorms, uninorms and nullnorms (t-operators) are special cases of n-uninorms. We also show that given n there are n + 1 classes of operators in U n and each of them has many subclasses. We also study the Frank equation involving n-uninorms and show that we need to consider only n-uninorms for the study. Finally, we show that the total number of subclasses of operators in U n follows the famous series called Catalan Numbers.

K- lp-Lipschitz t-norms

k- l p -Lipschitz t-norms are shown to be ordinal sums of k- l p -Lipschitz Archimedean t-norms. Additive generators of k- l p -Lipschitz t-norms are characterized by means of k- p-convexity. Several necessary and sufficient conditions for a function to be a k- p-convex additive generator are also given.

On the convex combination of TD and continuous triangular norms

The problem of when T λ ≜ ( 1 - λ ) T D + λ T λ ∈ ( 0 , 1 ) is a triangular norm, where T D is the drastic product and T is a continuous triangular norm, is studied. It is shown that T λ cannot be a triangular norm when T is nilpotent. It is also shown that T λ is a triangular norm if T is strict and its additive generator f satisfies f ( λ x ) = f ( x ) + f ( λ ) for all x ∈ [ 0 , 1 ] . The cases that T = T M and T is the ordinal sum of continuous Archimedean summands are also discussed. Some left-continuous t-norms which can be combined with each other are given.

Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Łukasiewicz logic and more recent results for Gödel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants { r ¯ | r ∈ C } , for a suitable countable C ⊆ [ 0 , 1 ] , and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Łukasiewicz, Gödel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Łukasiewicz components behave as rational numbers.

A structure theorem for posets admitting a “strong” chain partition:: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

Suppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q ∈ P lie in distinct chains and p < q , then every other element of P is either above p or below q. In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets ⊕ i = 1 n R i such that, for 1 ⩽ i ⩽ n , one of the following occurs: (1) R i is disjoint from one of the chains of the partition; or (2) if p, q ∈ R i are in distinct chains, then they are incomparable. The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets. In 1996, a proof of the Daykin–Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified. In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained.

On the structure of generalized BL-algebras

A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities $$ x\Lambda y = ((x\Lambda y)/y)y = y(y\backslash (x\Lambda y)) $$ . It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of $$ {\ell } $$ -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras.

On Weakly Cancellative Fuzzy Logics

Starting from a decomposition result of monoidal t-norm-based logic (MTL)-chains as ordinal sums, we focus our attention on a particular kind of indecomposable semihoops, namely weakly cancellative semihoops. The weak cancellation property is proved to be the difference between cancellation and pseudocomplementation, so it gives a new axiomatization of product logic and ΠMTL. By adding this property, some new fuzzy logics (propositional and first-order) are defined and studied obtaining some results about their (finite) strong standard completeness and other logical and algebraic properties.

Free algebras in varieties of BL-algebras generated by a BLn-chain

AbstractFree algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.

PERFECT CODES ON SOME ORDERED SETS

Using the concept of codes on ordered sets introduced by Brualdi, Graves and Lawrence, we consider perfect codes on the ordinal sum of two ordered sets, the standard ordered sets and the disjoint sum of two chains.

Every Linear Pseudo BL-Algebra Admits a State

We show that every linear pseudo BL-algebra, hence every representable one, admits a state and is good. This solves positively the problem on the existence of states raised in Dvurecenskij and Rachůnek (Probabilistic averaging in bounded communitative residuated l-monoids, 2006), and gives a partial answer to the problem on good pseudo BL-algebras from [Di Nola, Georgescu and Iorgulescu (Multiple Val Logic 8:715–750, 2002) Problem 3.21]. Moreover, we present that every saturated linear pseudo BL-algebra can be expressed as an ordinal sum of Hajek’s type of irreducible pseudo linear pseudo BL-algebras.

Distributivity and conditional distributivity of a uninorm and a continuous t-conorm

The open problem recalled by Klement in the Linz2000 closing session, related to distributivity and conditional distributivity of a uninorm and a continuous t-conorm, is solved for the most usual known classes of uninorms. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent for these cases. It is remarkable that solutions appear involving not only strict t-conorms but also ordinal sums of the maximum with a strict t-conorm. Conversely, the distributivity of a t-conorm over a uninorm is also studied leading only to already known solutions. Moreover, the dual case of distributivity and conditional distributivity involving uninorms and continuous t-norms is also solved, proving again the equivalence of both kinds of distributivities.

On ordinal sums of triangular norms on bounded lattices

Ordinal sums have been introduced in many different contexts, e.g., for posets, semigroups, t-norms, copulas, aggregation operators, or quite recently for hoops. In this contribution, we focus on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets. Necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t-norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t-norms. By such also the structure of the underlying bounded lattice is investigated. Further, it is shown that up to trivial cases there are no ordinal sum t-norms on product lattices in general.

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison's ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre–Hilbert space. A Hilbert space of densities, whose logarithm is square–integrable, is obtained as the natural completion of the pre–Hilbert space.

Aggregation operators based on indistinguishability operators

This article gives a new approach to aggregating assuming that there is an indistinguishability operator or similarity defined on the universe of discourse. The very simple idea is that when we want to aggregate two values a and b we are looking for a value λ that is as similar to a as to b or, in a more logical language, the degrees of equivalence of λ with a and b must coincide. Interesting aggregation operators on the unit interval are obtained from natural indistinguishability operators associated to t-norms that are ordinal sums. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 857–873, 2006.

Signed higher-radix full-adder algorithm and implementation with current-mode multi-valued logic circuits

A novel algorithm for full-addition of two signed, higher-radix numbers is proposed and implemented by combining multi-valued logic min, max, literal and cyclic operators. Owing to disjoint terms involved, multi-valued logic min and max operators are replaced with ordinary transmission operation and sum, respectively. A multi-valued logic cyclic gate is designed by using a current-mode threshold circuit while the literal is realised by only voltage-mode switching circuits. The threshold circuit employed within the cyclic gate exhibits improved dynamic behaviour compared to its previous counterparts employing voltage-mode binary logic switching circuits. It also allows much higher radices compared to previous current-mode threshold circuits owing to its superior mismatch properties. Thus, the cyclic gate achieves a superior performance compared to its predecessors. As a direct extension to cyclic operation in radix-8, a resultant single-digit, radix-8 full-adder and its 3-bit counterpart voltage-mode circuits are designed and their performance compared. It is shown that the developed signed addition algorithm can be realised by using the proposed full-adder. Finally, the algorithm is also exploited for a multi-digit case. Simulation results demonstrate that proposed architectures can be used in high-performance arithmetic units.