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Let be the class of odd involutive even the notion of partial lex products is not sufficiently general. One more tweak is needed, a slightly even more complex construction, called partial sublex product, introduced here.

Legumes are able to meet their nitrogen need by establishing nitrogen-fixing symbiosis with rhizobia. Nitrogen fixation is performed by rhizobia, which has been converted to bacteroids, in newly formed organs, the root nodules. In the model legume

Summary BACKGROUND & AIMS Although conclusive evodence is yet lacking, it has been suggested that vitamin D deficiency (VD) may be associated with a more severe course of SARS-CoV-2 Infection (COVID-19). In this retrospective study we assessed the association of VD deficiency with mortality in a group of COVID-19 patients treated in a tertiary referral center. METHODS Data of 257 Covid-19 patients hospitalized between 30th September 2020 and 2nd March 2021 have been collected retrospectively. The following parameters were collected: age, gender, serum level of 25-OH-Vitamin D3, outcome (survival/death) comorbidities (cancer, diabetes mellitus and chronic obstructive pulmonary disease). Serum VD measurement was done within 3 days of admission. RESULTS VD levels were significantly lower in patients who did not survive, however, in this patients’ group the average age was significantly higher than among those, who survived. After age-matching, in a subgroup of patients with risk factors and/or 60 years of age or older who survived had significantly higher VD level in their serum than those who deceased. Serum C-reactive protein, lactate-dehydrogenase and creatinin-kinase were significantly higher in the group in which the patients died, however these laboratory parameters did not correlate with the VD levels. CONCLUSION We found that in COVID-19 infection, when old age as risk factor (60 years of age or older) was pooled with risk factors (cancer, diabetes and/or COPD), the VD levels were significantly lower in the patient group, in which the patients did not survive. We suggest further, prospective studies in similar subgroups to explore a possible causal relationship.

Legumes of the Medicago genus form symbiosis with the bacterium Sinorhizobium meliloti and develop root nodules housing large numbers of the intracellular symbionts. Members of the Nodule-specific Cysteine Rich peptide (NCRs) family induce the endosymbionts into a terminal differentiated state. Individual cationic NCRs are antimicrobial peptides that have the capacity to kill the symbiont but the nodule cell environment prevents killing. Moreover, the bacterial broad-specificity peptide uptake transporter BacA and exopolysaccharides contribute to protect the endosymbionts against the toxic activity of NCRs. Here, we show that other S. meliloti functions participate in the protection of the endosymbionts, including an additional broad-specificity peptide uptake transporter encoded by the yejABEF genes, lipopolysaccharide modifications mediated by lpsB and lpxXL as well as rpoH1, encoding a stress sigma factor. Mutants of these genes show in vitro a strain-specific increased sensitivity profile against a panel of NCRs and form nodules in which bacteroid differentiation is affected. The lpsB mutant nodule bacteria do not differentiate, the lpxXL and rpoH1 mutants form some seemingly fully differentiated bacteroids although most of the nodule bacteria are undifferentiated, while the yejABEF mutants form hypertrophied but nitrogen-fixing bacteroids. The nodule bacteria of all the mutants have a strongly enhanced membrane permeability, which is dependent on the transport of NCRs to the endosymbionts. Our results suggest that S. meliloti relies on a suite of functions including peptide transporters, the bacterial envelope structures and stress response regulators to resist the aggressive assault of NCR peptides in the nodule cells.ImportanceThe nitrogen fixing symbiosis of legumes with rhizobium bacteria has a predominant ecological role in the nitrogen cycle and has the potential to provide the nitrogen required for plant growth in agriculture. The host plants allow the rhizobia to colonize specific symbiotic organs, the nodules, in large numbers in order to produce sufficient reduced nitrogen for the plant needs. Some legumes, including Medicago spp., produce massively antimicrobial peptides to keep this large bacterial population in check. These peptides, known as NCRs, have the potential to kill the rhizobia but in nodules, they rather inhibit the division of the bacteria, which maintain a high nitrogen fixing activity. In this study, we show that the tempering of the antimicrobial activity of the NCR peptides in the Medicago symbiont Sinorhizobium meliloti is multifactorial and requires the YejABEF peptide transporter, the lipopolysaccharide outer membrane composition and the stress response regulator RpoH1.

Uninorms play a prominent role both in the theory and the applications of Aggregations and Fuzzy Logic. In this paper the class of group-like uninorms is introduced and characterized. First, two variants of a general construction -- called partial-lexicographic product -- will be recalled from \cite{Jenei_Hahn}; these construct odd involutive FL$_e$-algebras. Then two particular ways of applying the partial-lexicographic product construction will be specified. The first method constructs, starting from $\mathbb R$ (the additive group of the reals) and modifying it in some way by $\mathbb Z$'s (the additive group of the integers), what we call basic group-like uninorms, whereas with the second method one can modify any group-like uninorm by a basic group-like uninorm to obtain another group-like uninorm. All group-like uninorms obtained this way have finitely many idempotent elements. On the other hand, we prove that given any group-like uninorm which has finitely many idempotent elements, it can be constructed by consecutive applications of the second construction (finitely many times) using only basic group-like uninorms as building blocks. Hence any basic group-like uninorm can be built using the first method, and any group-like uninorm which has finitely many idempotent elements can be built using the second method from only basic group-like uninorms. In this way a complete characterization for group-like uninorms which possess finitely many idempotent elements is given: ultimately, all such uninorms can be built from $\mathbb R$ and $\mathbb Z$. This characterization provides, for potential applications in several fields of fuzzy theory or aggregation theory, the whole spectrum of choice of those group-like uninorms which possess finitely many idempotent elements.

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL , posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL$$_e$$ e -chains which have finitely many positive idempotent elements.

Uninorms play a prominent role both in the theory and applications of Aggregations and of Mathematical Fuzzy Logic. In this paper the class of group-like uninorms is introduced as the subclass of group-like FL\(_e\)-algebras, where the underlying universe is order-isomorphic to the open real unit interval. First, two variants of a construction method—called partial-lexicographic product—will be recalled; it constructs a large subclass of group-like FL\(_e\)-algebras. Then three specific ways of applying the partial-lexicographic product construction will be introduced. The first one constructs starting from \(\mathbb R\) and modifying it in some way by \(\mathbb Z\)’s, what we call the basic group-like uninorms, whereas with the last two ones one may extend group-like uninorms by using \(\mathbb Z\) and basic uninorms to obtain further group-like uninorms. All group-like uninorms obtained this way have finitely many idempotents. On the other hand, we assert that the only way to construct group-like uninorms which have finitely many idempotents is to apply the last two extension methods consecutively, starting from a basic group-like uninorm. In this way a complete characterization for group-like uninorms which possess finitely many idempotents is given. The paper is illustrated with several 3D plots of group-like uninorms.

For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunn about finite Sugihara monoids., Comment: This is the version which appeared in Studia Logica

Hahn’s celebrated embedding theorem asserts that linearly ordered Abelian groups embed in the lexicographic product of real groups [13]. In this paper the partial-lexicographic product construction is introduced, a class of residuated monoids, namely, group-like FL\(_e\)-chains which possess finitely many idempotents are embedded into finite partial-lexicographic products of linearly ordered Abelian groups, that is, Hahn’s theorem is extended to this residuated monoid class. As a side-result, the finite strong standard completeness of the logic \(\mathbf{IUL}^{fp}\) is announced.

A construction method, called connected co-rotation, is introduced in this paper. The precursor of its disconnected version has been introduced by R.K. Meyer. Both co-rotation constructions result in residuated semigroups, among them non-positive rank involutive FL e -algebras. These constructions can be considered as ‘skew-dual’ to their rotation counterparts (those without the co-syllable). The classes of residuated semigroups, for which the connected co-rotation construction results in associative operations, are characterized. Since the constructions are suitable to construct non-positive rank involutive FL e -algebras, co-rotations provide a better insight into the algebraic semantic of the substructural logic IUL.

Classification of certain group-like FL $$_e$$ -chains is given: We define absorbent-continuity of FL $$_e$$ -algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL $$_e$$ -algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, and with no two consecutive cancellative components. It is shown that the classification theorem does not hold if we drop the absorbent-continuity condition. Our result is the first classification theorem in the literature on FL $$_e$$ -algebras that does not assume the condition of being naturally ordered (which, under certain conditions, corresponds to continuity of the monoid operation). In our classification theorem, continuity is replaced by the much weaker absorbent-continuity.

A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabzinski and Wronski: we provide an equational characterization for the equivalence operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety.

The paper deals with involutive FL e -monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FL e -monoids over lattices are exactly involutive FL e -algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FL e -monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FL e -chains are classified by using the notion of rank of involutive FL e -chains, and a kind of duality is developed between positive and non-positive rank algebras. As a side effect, it is shown that the substructural logic IUL plus t ? f does not have the finite model property.

We investigate uninorm algebras satisfying a strong version of involutiveness. More precisely, we require that negation is an order reversing monoid isomorphism between the positive cone and the negative cone. A rather surprising consequence of this property is that the negative cones of these algebras are BL-algebras which do not admit MV-components with more than two elements. Among other things, we prove standard completeness and co-NP completeness of the logic corresponding to these algebras.

By analogy with the usual extension of the group operation from the positive cone of an ordered Abelian group into the whole group, a construction-called symmetrization-is defined and it is related to the rotation construction [Jenei, On the structure of rotation-invariant semigroups, Archive for Mathematical Logic 42 (2003) 489-514]. Symmetrization turns out to be a kind of dualized rotation. A characterization is given for the left-continuous t-conorms for which their symmetrization is a uninorm. As a by-product a new family of involutive uninorms is introduced.

It is shown that, under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. Finally, these results are applied for the subdomains of uniqueness problem, yielding new results.

The main ‘philosophical’ outcome of this article is to demonstrate that the structural description of residuated lattices requires the use of the co-residuated setting. A construction, called skew symmetrization, which generalizes the well-known representation of an ordered Abelian group obtained from the positive (or negative) cone of the algebra is introduced here. Its definition requires leaving the accustomed residuated setting and entering the co-residuated setting. It is shown that every uninorm on [0, 1] with an involution defined by the residual complement with respect to the unit and having the unit as the fixed point of the involution can be described as the skew symmetrization of its underlying t-norm or underlying t-conorm.

A geometric characterization of residuated semigroups is presented in this paper based on the notions of the rotation-invariance property, and the pseudo-inverse of monotone functions. This provides a deeper understanding of associativity via geometric insight. It is then demonstrated how intuition, which is gained from the geometric description may help in obtaining results in the fields of algebra, logic and functional equations.

A conjecture of C. Alsina, M. J. Frank and B. Schweizer concerning the convex combinations of t-norms [4] is proved for certain left-continuous t-norms. It is shown that a nontrivial convex combination of two left-continuous t-norms is never a t-norm (in fact, the associativity property of t-norms is violated) provided that the two t-norms have the same involutive u-level set for some u∈[0, 1[. The proof is motivated by a geometrical understanding of associativity.

The question of determining uniquely either a continuous Archimedean t-norm or a left-continuous t-norm on some vertical segments and/or on some level sets of its graph is investigated.

In this paper, it is shown that aggregation operators cannot be associative and strictly compensatory at the same time. As a way out, the rotation construction and the rotation-annihilation construction are applied to construct wide classes of associative aggregation operators that are compensatory on a subdomain only. From this point of view, the resulting operators are similar to uninorms, a class of aggregation operators that is being reported on extensively in the literature. The constructions are demonstrated by means of ample examples.

Left-continuity of triangular norms is the characteristic property to make it a residuated lattice. Nowadays residuated lattices are subjects of intense investigation in the fields of universal algebra and nonclassical logic. The recently known construction methods resulting in left-continuous triangular norms are surveyed in this paper.

Left-continuous t-norms are much more complicated than the continuous ones, and obtaining a complete classification of them seems to be a very hard task. In this paper we investigate some aspects of left-continuous t-norms, with emphasis on their continuity points. In particular, we are interested in left-continuous t-norms which are isomorphic to t-norms which are continuous in the rationals. We characterize such a class, and we prove that it contains the class of all weakly cancellative left-continuous t-norms.

In this paper, a method is presented for constructing t-norms on product lattices (in other words: commutative partially ordered integral monoids over product lattices) which are not direct products. The method is fairly general and allows to generate a broad class of such t-norms. This solves an open problem posed in 1999 by De Baets and Mesiar.

We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms.

A new method for constructing (left-continuous) t-norms is introduced and analyzed in this paper. We shall construct via embedding a left-continuous t-norm from any countable residuated totally and densely ordered commutative integral monoid. Moreover, we can construct a left-continuous t-norm from any countable, totally ordered, commutative integral monoid which is not necessarily densely ordered and residuated. A special case, the embedding of such monoids on lexicographic product spaces is investigated in detail, and several examples are demonstrated. The results shed some light to Chang's MV-algebra, to a recently proposed 'extraordinary' t-norm, and to the standard semantics of the recently introduced logic II-MTL of Hajek.

We introduce the notion of a dice model as a framework for describing a class of probabilistic relations. We investigate the transitivity of the probabilistic relation generated by a dice model and prove that it is a special type of cycle-transitivity that is situated between moderate stochastic transitivity or product-transitivity on the one side, and Łukasiewicz-transitivity on the other side. Finally, it is shown that any probabilistic relation with rational elements on a three-dimensional space of alternatives which possesses this particular type of cycle-transitivity, can be represented by a dice model. The same does not hold in higher dimensions.

This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. The interpolation/extrapolation method which was proposed for one-dimensional input space in [4] is extended in this paper to the general n-dimensional case by using the concept of aggregation operators. A characterization of the class of aggregation operators with which the extended method preserves all the nice features of the one- dimensional method is given.

In this paper, the well-known ordinal sum theorem of semigroups is generalized and applied to construct new families of triangular subnorms and triangular norms (t-norms). Among them one can find several new families of left-continuous t-norms too.

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the involutive monoidal t-norm-based residuated logic.

First, a characterization is given for the continuity of a class of left-continuous triangular norms: it is established that a left-continuous triangular norm with strong induced negation is continuous if and only if it is strictly increasing on the domain where its value is positive. Second, we show that axioms of a left-continuous triangular norm with strong induced negation are not independent, namely, the boundary condition follows from the other axioms.