We show that the equational theory of representable lattice-ordered residuated semigroups is not finitely axiomatizable. We apply this result to the problem of completeness of substructural logics.

We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics.

Hahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for (...) odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented. (shrink)

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.

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.

A Correction to this paper has been published: 10.1007/s11225-020-09933-y.

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by very (...) simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them. (shrink)

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the (...) Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models. (shrink)

The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay, law. The Wigner–Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does riot correspond to a semigroup (...) (experiments on the decay of the neutral K-meson system strongly support the semigroup evolution postulated, by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax–Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax–Phillips S-matrix is an inner function (the Lax–Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the S-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle (V, N, θ) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state. (shrink)

A many-valued sentential logic with truth values in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV-algebra. In the well-defined fuzzy sentential (...) logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. (shrink)

In this book the notion of semigroups under + is constructed using: the MOD natural neutrosophic integers, or MOD natural neutrosophic-neutrosophic numbers, or MOD natural neutrosophic finite complex modulo integer, or MOD natural neutrosophic dual number integers, or MOD natural neutrosophic special dual like number, or MOD natural neutrosophic special quasi dual numbers.

Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG of complete types concentrated on G is an inverse limit of the ∞-definable semigroups SG,Δ. He also showed that it is strongly π-regular: for every p∈SG,Δ, there exists n∈N such that pn is in a subgroup of SG,Δ. We show that SG,Δ is in fact an intersection of definable semigroups, so SG is an inverse limit of definable semigroups, and (...) that the latter property is enjoyed by all ∞-definable semigroups in stable structures. (shrink)

Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable.

A system of semigroup identities is hereditarily finitely based if it defines a variety all semigroups of which are finitely based. Two new types of hereditarily finitely based identity systems are presented. Two of these systems, together with eight existing systems, establish the hereditary finite basis property of every semigroup of order five or less with one possible exception.

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. (...) However, for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich’s erroneous contribution in (...) this area, we present A. I. Maltsev’s example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally. (shrink)

We prove a representation theorem for residuated algebras: each residuated algebra is isomorphically embeddable into a powerset residuated algebra. As a consequence, we obtain a completeness theorem for the Generalized Lambek Calculus. We use a Labelled Deductive System which generalizes the one used by Buszkowski [4] and Pankrat'ev [17] in completeness theorems for the Lambek Calculus.

This is also where we begin investigating lattices of logics and varieties, rather than particular examples.

Two approaches toward the arrow of time for scattering processes have been proposed in rigged Hilbert space quantum mechanics. One, due to Arno Bohm, involves preparations and registrations in laboratory operations and results in two semigroups oriented in the forward direction of time. The other, employed by the Brussels-Austin group, is more general, involving excitations and de-excitations of systems, and apparently results in two semigroups oriented in opposite directions of time. It turns out that these two time arrows (...) can be related to each other via Wigner's extensions of the spacetime symmetry group. Furthermore, their are subtle differences in causality as well as the possibilities for the existence and creation of time-reversed states depending on which time arrow is chosen. (shrink)

We discuss issues related to constructing an orthomodular structure from an object in a category. In particular, we consider axiomatics related to Baer *-semigroups, partial semigroups, and various constructions involving dagger categories, kernels, and biproducts.

We share Nair-Collins and Joffe's (2023) concern with the accuracy of the “whole brain-death” diagnosis, which fails to take into account current understandings of residual brain function (neurohor...

The variety of residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., \-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated \-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated \-groupoids, their ideals, and develop a theory of left nuclei. (...) Finally, we extend some parts of the theory of join-completions of residuated \-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices. (shrink)

We present an algebraic approach to canonical embeddings of arbitrary residuated algebras into powerset residuated algebras. We propose some construction of powerset residuated algebras and prove a representation theorem for symmetric residuated algebras.

Neutrosophic LA-semigroup is a midway structure between a neutrosophic groupoid and a commutative neutrosophic semigroup. Rings are the old concept in algebraic structures. We combine the neutrosophic LA-semigroup and ring together to form the notion of neutrosophic LA-semigroup ring. Neutrosophic LAsemigroup ring is defined analogously to neutrosophic group ring and neutrosophic semigroup ring.

In this paper we extend the theory of neutrosophy to study left almost semigroup shortly LAsemigroup. We generalize the concepts of LA-semigroup to form that for neutrosophic LA-semigroup. We also extend the ideal theory of LA-semigroup to neutrosophy and discuss different kinds of neutrosophic ideals. We also find some new type of neutrosophic ideal which is related to the strong or pure part of neutrosophy. We have given many examples to illustrate the theory of neutrosophic LA-semigroup and display many properties (...) of neutrosophic LA-semigroup in this paper. (shrink)

Molodtsov introduced 1999 the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to ordered semigroups. The notions of soft ordered semigroup, soft ordered subsemigroup, soft left ideal, and left idealistic soft ordered semigroup are introduced, and various related properties are investigated.

This essay reflects on the concept of “hydropower” – the corrosive power of seawater to amalgamate Life and Nonlife in the context of migrant deaths in the waters of the Mediterranean. Through a focus on drowned bodies’ dissolution and eventual sedimentation into their deep-sea surroundings, my approach interrelates the order of biopolitical violence enacted by Europe’s restrictive migration policies and the thick time of the geophysical. The degradation of bodies under the influence of hydropower reveals residual ontologies marked by porousness (...) between embodied forms of Life and their geophysical environments, putting significant pressure on the putatively watertight divide between Life and Nonlife in the Anthropocene. Parsed from the lens of residuality, hydropower reveals humans’ full ontological coincidence with matter writ large, their endurance and solubility in geological life forces, but also the necessity to think agency in terms of human/inhuman continuity in excess of biopower’s regimenting forces. Against the attempted biopolitical suppression of a certain form of humanity, the residual dwelling enacted by hydropower champions the inclusion of new constellations of matter in our political thought processes. (shrink)

We introduce a new product bilattice con- struction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lat- tice factors. Finally, we (...) show how to employ our product construction to define first-order definable classes of bi- lattices corresponding to any first-order definable subclass of residuated lattices. (shrink)

Residual categories such as »not elsewhere categorized« densely populate modern information systems. This article roughly categories two types of modern information surveillance and notification systems, statistical and event-based. It examines the nature of residual categories arising from each, and proposes some methodological considerations for how these impact moral order within information infrastructure. The article concludes with comments about how the inclusion of lived experience might ameliorate a sort of moral gridlock often encountered today in large-scale information systems.

This paper will examine the concepts of integrity and moral residue as they relate to nursing practice in the current health care environment. I will begin with my definition and conception of ethical practice, and, based on that, will go on to argue for the importance of recognizing that nurses often find themselves in the position of compromising their moral integrity in order to maintain their self‐survival in the hospital or health care environment. I will argue that moral integrity is (...) necessary to a moral life, and is relational in nature. When integrity is threatened, the result is moral distress, moral residue, and in some cases, abandonment of the profession. The solution will require more than teaching bioethics to nursing students and nurses. It will require changes in the health care environment, organizational culture and the education of nurses, with an emphasis on building a moral community as an environment in which to practise ethically. (shrink)

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

A propositional framework of formal reasoning is proposed, which emphasises the pattern of entering and exiting context. Contexts are modelled by an algebraic structure which reflects the order and manner in which context is entered into and exited from.The equations of the algebra partitions context terms into equivalence classes. A formal semantics is defined, containing models that map equivalence classes of certain context terms to sets of interpretations of the formula language. The corresponding Hilbert system incorporates the algebraic equations as (...) axioms asserted in context.In semigroups of contexts, where combination of contexts is associative, finite ground algebraic equations correspond to contingent equivalence between certain logical formulas. Systems for sets and multisets of contexts are obtained by presenting their respective algebras as associativity plus finite ground equations. Soundness and completeness results are proved.Some contextual reasoning systems in the literature are inherently associative, and we present those as special cases. (shrink)

In this paper we define neutrosophic bi-LAsemigroup and neutrosophic N-LA-semigroup. Infact this paper is an extension of our previous paper neutrosophic left almost semigroup shortly neutrosophic LAsemigroup. We also extend the neutrosophic ideal to neutrosophic biideal and neutrosophic N-ideal. We also find some new type of neutrosophic ideal which is related to the strong or pure part of neutrosophy. We have given sufficient amount of examples to illustrate the theory of neutrosophic bi-LA-semigroup, neutrosophic N-LAsemigroup and display many properties of them (...) this paper. (shrink)

Soft set theory is a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. In this paper we introduced soft neutrosophic biLA-semigroup,soft neutosophic sub bi-LA-semigroup, soft neutrosophic N -LA-semigroup with the discuission of some of their characteristics. We also introduced a new type of soft neutrophic bi-LAsemigroup, the so called soft strong neutrosophic bi-LAsemigoup which is of pure neutrosophic character. This is also extend to soft neutrosophic strong N-LA-semigroup. We also given some of their properties of this (...) newly born soft structure related to the strong part of neutrosophic theory. (shrink)

We show (1) the consequence determined by a variety V of algebraic semigroup matrices is finitely based iffV is finitely based, (2) the consequence determined by all 2-valued semigroup connectives, Λ, ∨, ↔, +, in other words the collection of common rules for all these connectives, is finitely based. For possible applications see Sect. 0.

One of the debated issues regarding Residual Normality (RN) is frequency sensitivity in Williams syndrome (WS). We present some data on frequency sensitivity in Hungarian WS subjects. Based on vocabulary measures, we suggest that instead of the across-the-board frequency insensitivity proposed by some, a higher frequency threshold characterizes these subjects’performance. Results from a category fluency task show that whereas frequency sensitivity in WS is in line with controls, error patterns imply a qualitatively distinct, looser categorical organization. Regarding the much-debated issue (...) of morphological overgeneralizations, our data suggest that frequency sensitivity cuts across the divisions proposed by dual-process theories. In general, some of the frequency effects are the same as in typically developing populations, but with a delayed pattern. Frequency may be interpreted as supporting RN, but in WS it operates with higher thresholds that might be a general processing feature of WS individuals. (shrink)

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex (...) theory. (shrink)

Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we consider pairs (...) consisting of a Brouwerian algebra and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s-equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL-algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV-algebras. (shrink)

A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.

This article focuses on three scenarios in which residual biological materials are turned into research collections during the procedure of procuring these materials for diagnostic, therapeutic or other non-research purposes. These three scenarios differ from each other primarily because they employ different models of consent: (a) precautionary consent, which may be secured during the collecting procedure; (b) the presumed consent model, which may be applied during the collection of materials; and (c) consent for research use of identifiable human biological materials, (...) which may be skipped entirely. These scenarios offer additional sources of biological samples for research purposes and at the same time seem to offer even more flexibility in terms of stringency of consent as compared with the more traditional models of broad consent in prospective research collections and the waiver of consent in retrospective research. Our discussion leads us to think that precautionary consent is preferable to presumed consent and no consent when handling issues of consent in the use of residual human biological materials for research. However, such precautionary consent should not be construed as blanket, unrestricted consent for any future use. (shrink)

Residual useful lifetime prediction plays a key role of failure prediction and health management in equipment. Aiming at the problems of residual life prediction without comprehensively considering multistage and individual differences in equipment performance degradation at present, we explore a prediction model that can fit the multistage random performance degradation. Degradation modeling is based on the random Wiener process. Moreover, according to the degradation monitoring data of the same batch of equipment, we apply the expectation maximization algorithm to estimate the (...) prior distribution of the model. The real-time remaining life distribution of the equipment is acquired by merging prior information of real-time degradation data and historical degradation monitoring data. The accuracy of the proposed model is demonstrated by analyzing a practical case of metalized film capacitors, and the result shows that a better RUL estimation accuracy can be provided by our model compared with existing models. (shrink)

Patients with severe disorders of consciousness are thought to be unaware of themselves or their environment. However, research suggests that a minority of patients diagnosed as having a disorder of consciousness remain aware. These patients, designated as having “cognitive motor dissociation”, can demonstrate awareness by imagining specific tasks, which generates brain activity detectable via functional neuroimaging. The discovery of consciousness in these patients raises difficult questions about their well-being, and it has been argued that it would be better for these (...) patients if they were allowed to die. Conversely, I argue that CMD patients may have a much higher level of well-being than is generally acknowledged. It is far from clear that their lives are not worth living, because there are still significant gaps in our understanding of how these patients experience the world. I attempt to fill these gaps, by analyzing the neuroscientific research that has taken place with these patients to date. Having generated as comprehensive a picture as possible of the capacities of CMD patients, I examine this picture through the lens of traditional philosophical theories of well-being. I conclude that the presumption that CMD patients do not have lives worth living is not adequately supported. (shrink)

Starting with a partial order on a NeutroAlgebra, we get a NeutroStructure. The latter if it satisfies the conditions of NeutroOrder, it becomes a NeutroOrderedAlgebra. In this paper, we apply our new defined notion to semigroups by studying NeutroOrderedSemigroups. More precisely, we define some related terms like NeutrosOrderedSemigroup, NeutroOrderedIdeal, NeutroOrderedFilter, NeutroOrderedHomomorphism, etc., illustrate them via some examples, and study some of their properties.

We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism of the subsemigroups yields isomorphism of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a computable structure, up to computable isomorphism, from the isomorphism type of its inverse semigroup (...) of computable partial automorphisms. (shrink)

In this arresting book, Wai Chee Dimock takes on the philosophical tradition from Kant to Rawls, challenging its conception of justice as foundational, self-evident, and all-encompassing. The idea of justice is based on the premise that the world can be resolved into commensurate terms: punishment equal to the crime, redress equal to the injury, benefit equal to the desert. Dimock focuses, however, on what remains unexhausted, unrecovered, and noncorresponding in the exercise of justice. To honor these "residues," she turns to (...) literature, which, in its linguistic density, transposes the clean abstractions of law and philosophy into persistent shadows, the abiding presence of the incommensurate. Justice can only be a partial answer to the phenomenon of human conflict. In arguing for justice as an _incomplete_ virtue, Dimock draws upon legal history, political philosophy, linguistics, theology, and feminist theory; she discusses Aristotle and Augustine, Locke and Luther, Marx and Durkheim, Michael Sandel and Carol Gilligan, Noam Chomsky and Mary Ann Glendon. She also examines an unusual configuration of nineteenth-century American authors, pairing figures such as Herman Melville and Rebecca Harding Davis, Walt Whitman and Susan Warner. The result is a book both passionate and scholarly. It invites us to rethink the meanings of literature, law, and philosophy, and to imagine a language of community more supple and more nuanced than the language of justice. (shrink)

In this arresting book, Wai Chee Dimock takes on the philosophical tradition from Kant to Rawls, challenging its conception of justice as foundational, self-evident, and all-encompassing. The idea of justice is based on the premise that the world can be resolved into commensurate terms: punishment equal to the crime, redress equal to the injury, benefit equal to the desert. Dimock focuses, however, on what remains unexhausted, unrecovered, and noncorresponding in the exercise of justice. To honor these "residues," she turns to (...) literature, which, in its linguistic density, transposes the clean abstractions of law and philosophy into persistent shadows, the abiding presence of the incommensurate. Justice can only be a partial answer to the phenomenon of human conflict. In arguing for justice as an _incomplete_ virtue, Dimock draws upon legal history, political philosophy, linguistics, theology, and feminist theory; she discusses Aristotle and Augustine, Locke and Luther, Marx and Durkheim, Michael Sandel and Carol Gilligan, Noam Chomsky and Mary Ann Glendon. She also examines an unusual configuration of nineteenth-century American authors, pairing figures such as Herman Melville and Rebecca Harding Davis, Walt Whitman and Susan Warner. The result is a book both passionate and scholarly. It invites us to rethink the meanings of literature, law, and philosophy, and to imagine a language of community more supple and more nuanced than the language of justice. (shrink)