Łukasiewicz Tribes are Absolutely Sequentially Closed Bold Algebras

We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean MV-algebra) can be uniquely extended to a sequentially continuous measure on the generated ukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on ukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields a characterization of the ukasiewicz tribes as bold algebras absolutely sequentially closed with respect to the extension of probabilities. The result generalizes the relationship between fields of sets and the generated -fields discovered by J. Novák. We introduce the category of bold algebras and sequentially continuous homomorphisms and prove that ukasiewicz tribes form an epireflective subcategory. The restriction to fields of sets yields the epireflective subcategory of -fields of sets.     

Universal Properties of Łukasiewicz Consequence

Boolean logic deals with {0, 1}-observables and yes–no events, as many-valued logic does for continuous ones. Since every measurement has an error, continuity ensures that small measurement errors on elementary observables have small effects on compound observables. Continuity is irrelevant for {0, 1}-observables. Functional completeness no longer holds when n-ary connectives are understood as [0, 1]-valued maps defined on [0, 1] ⁿ . So one must envisage suitable selection criteria for [0, 1]-connectives. ?ukasiewicz implication has a well known characterization as the only continuous connective ${\Rightarrow}$ satisfying the following conditions: (i) ${x\Rightarrow(y\Rightarrow z)= y\Rightarrow(x\Rightarrow z)}$ and (ii) ${x\Rightarrow y=1\, \, {\rm iff} x\leq y}$ . Then syntactic consequence can be defined purely algorithmically using the ?ukasiewicz axioms and Modus Ponens. As discussed in this paper, to recover a strongly complete semantics one may use differential valuations.     

Product Ł ukasiewicz Logic

u logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of u logic by adding a new connective which expresses multiplication. The resulting logic, P, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of P logic is introduced and developed too.The work of the first author was supported by the Grant Agency of the Czech Republic under project GACR 201/02/1540, by the Grant Agency of the Czech Technical University in Prague under project CTU 0208613, and by Net CEEPUS SK-042.The work of the second author was supported by grant IAA1030004 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

Reverse triple I reasoning methods based on the Łukasiewicz implication

The reverse and α-reverse triple I reasoning methods based on ?ukasiewicz implication I _L are established in new manners which correct the mistakes in the existing literature. Furthermore, the α-reverse triple I reasoning method is extended to a new form, called α(u, ν)-reverse triple I reasoning method, which can contain the reverse triple I reasoning method as its particular case. This is another improved point to the existing results.     

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution.     

A characterization of tribes with respect to the Łukasiewicz t-norm

We give a complete characterization of tribes with respect to the ukasiewicz t-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the ukasiewicz t-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental t-norms, e. g., for the product t-norm.     

States in Łukasiewicz logic correspond to probabilities of rational polyhedra

It will be shown that probabilities of infinite-valued events represented by formulas in ?ukasiewicz propositional logic are in one-to-one correspondence with tight probability measures over rational polyhedra in the unit hypercube. This result generalizes a recent work on rational measures of polyhedra and provides an elementary geometric approach to reasoning under uncertainty with states in ?ukasiewicz logic.     

Comprehension contradicts to the induction within Łukasiewicz predicate logic

We introduce the simpler and shorter proof of Hajek’s theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within ?ukasiewicz predicate logic ? ${\forall}$ (Hajek Arch Math Logic 44(6):763–782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra.     

Ringel–Hall Algebras of Quotient Algebras of Path Algebras

We determine the generating relations for Ringel–Hall algebras associated with quotient algebras of path algebras of Dynkin and tame quivers, and investigate their connection with composition subalgebras.     

Pseudomonotonicity of Nonlinear Transformations on Euclidean Jordan Algebras

In this paper, we have introduced the concepts of pseudomonotonicity properties for nonlinear transformations defined on Euclidean Jordan algebras. The implications between this property and other P-properties have been studied. More importantly, we have solved the solvability problem of the nonlinear pseudomonotone complementarity problems over symmetric cones.     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

Automorphism Group of Toeplitz Algebras on Certain Pseudoconvex Domains

in this paper, we characterize the automorphism groups of Toeplitz algebras oncertain strongly pseudoconvex domains of Cn, and obtain a generalized BDF theorem for aspecial kind of essential normal operator tupies.     

Characterizations of Additive ��-Lie Derivations on Unital Algebras

Ukrainian Mathematical Journal - Let R be a commutative ring with unity and let ?? be a unital algebra over ? (or a field ??). An R-linear map L : ??...     

Algebras that satisfy Auslander’s condition on vanishing of cohomology

Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture—by a 2003 counterexample due to Jorgensen and Şega—motivates the consideration of the class of rings that do satisfy Auslander’s condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander–Reiten Conjecture is proved for AC rings, and Auslander’s G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.     

Rota-Baxter Operators of Weight 1 on 2×2-matrix Algebras

In this paper we determine all Rota-Baxter operators of weight 1 on 2×2-matrix algebras over an algebraically closed field.     

One-Point Extensions of t-Koszul Algebras

We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.     

A Note on Purely Imaginary Roots of Generalized Kac–Moody Algebras

Abstract

In this paper, we generalize the concept of purely imaginary roots of Kac–Moody algebras to generalized Kac–Moody algebras. Also we give a complete classification of those generalized Kac–Moody algebras with the purely imaginary property. We also define a new class of indefinite non-hyperbolic generalized Kac–Moody algebras called extended hyperbolic generalized Kac–Moody algebras and find that it does not always possess the purely imaginary property whereas the extended hyperbolic Kac–Moody algebras possess the purely imaginary property.