BL algebras were introduced by Ha Âjek as algebraic structures for his Basic Logic, starting from continuous t-norms on 0; 1. MV algebras, product algebras and Go Èdel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. We have generalized the BL algebras and pseudo-MV algebras, introducing the pseudo-BL algebras. In this paper we introduce weak-BL algebras and weak-pseudo-BL algebras. We also introduce non-commutative t-norms (we call them pseudo-t-norms) and use them in constructing pseudo-BL algebras and weak-pseudo-BL algebras. IntroductionWehave started from the following situation:(1) We know BL algebras, algebraic structures for which the principal examples come from: ± the real interval [0, 1] with the structure given by a continuous t-norm and ± abelian l-groups.(2) The non-commutative case (pseudo-MV algebras and pseudo-BL algebras) was developed starting from arbitrary l-groups.The natural problem was: can be de®ned a concept of pseudo-t-norm (by weaking the axioms of t-norms) on [0, 1] or, more general, on bounded chains, bounded lattices, in order to obtain new examples of pseudo-BL algebras?The present paper tries to answer to this problem in the following way:(1) We de®ne the notion of pseudo-t-norm, by throwing away the axiom of commutativity; (2) On arbitrary l-groups, we re®nd the already known examples;(3) On [0, 1], the condition of continuity is replaced by the weaker condition of left continuity in both variables. Then, [0, 1] is endowed with a weaker structure, that of weak-pseudo-BL algebra. This structure gives birth to the new concept of weak-BL algebra, in the commutative case.In conclusion, we de®ne a concept of pseudo-t-norm, that leads to pseudo-BL algebras on arbitrary l-groups; on [0, 1], we can not do this, without exiting from commutative case, the adequate structure being that of weakpseudo-BL algebra. In this paper we present a general up-today picture of the algebra of non-commutative logic. t-Norms and U-operatorsFirst, we shall recall the de®nitions of t-norms (t-conorms) and of their associated U-operators de®ned on the real interval [0, 1]. De®nition 2.1 (Cf. [32]) (a) A binary operation T on the real interval [0, 1] is a t-norm iff: (t0) it is commutative, (t1) it is associative, (t2) it is non-decreasing (isotone) in the ®rst argument (i.e. if x y, then Tx; z T y; z, for Focus All our gratitude to Radko Mesiar for his patient reading of the manuscript. We all three are new in the ®eld of t-norms, therefore his very kind suggestions were very useful to improve the paper, as for example: to complete the bibliography with [13, 14, 18, 41, 42], to modify the proof of Theorem 5.15; he noted that our Examples 8.3(1) is a left-continuous modi®cation of Example 1.12 from [42] and our further examples of pseudo-t-norms are ordinal sums in the sense of Clifford (Theorem 3.42 from [42] or [41]); he also noted that then relevant residual operators h...

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BackgroundThe response to endotoxin (LPS), and subsequent signal transduction lead to the production of cytokines such as tumor necrosis factor-α (TNF-α) by innate immune cells. Cells or organisms pretreated with endotoxin enter into a transient state of hyporesponsiveness, referred to as endotoxin tolerance (ET) which represents a particular case of negative preconditioning. Despite recent progress in understanding the molecular basis of ET, there is no consensus yet on the primary mechanism responsible for ET and for the more complex cases of cross tolerance. In this study, we examined the consequences of the macromolecular crowding (MMC) and of fractal-like kinetics (FLK) of intracellular enzymatic reactions on the LPS signaling machinery. We hypothesized that this particular type of enzyme kinetics may explain the development of ET phenomenon.MethodOur aim in the present study was to characterize the chemical kinetics framework in ET and determine whether fractal-like kinetics explains, at least in part, ET. We developed an ordinary differential equations (ODE) mathematical model that took into account the links between the MMC and the LPS signaling machinery leading to ET. We proposed that the intracellular fractal environment (MMC) contributes to ET and developed two mathematical models of enzyme kinetics: one based on Kopelman’s fractal-like kinetics framework and the other based on Savageau’s power law model.ResultsKopelman’s model provides a good image of the potential influence of a fractal intracellular environment (MMC) on ET. The Savageau power law model also partially explains ET. The computer simulations supported the hypothesis that MMC and FLK may play a role in ET.ConclusionThe model highlights the links between the organization of the intracellular environment, MMC and the LPS signaling machinery leading to ET. Our FLK-based model does not minimize the role of the numerous negative regulatory factors. It simply draws attention to the fact that macromolecular crowding can contribute significantly to the induction of ET by imposing geometric constrains and a particular chemical kinetic for the intracellular reactions.