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)

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)

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.

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)

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)

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.

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)

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...

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)

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.

Some contemporary theories about the origin and the nature of language resort to concepts with no bearing on Darwinian evolutionary hypothesis or evo-devo perspective which are both based on the reconstruction of species morphological structure transformation. These theories, which evoke qualitative leap, cultural evolution, structure/function coevolution as esplicative principles for human evolution, in our opinion, result compatible in some points with the most recent Intelligent Design accounts. Attempting to substantiate itself as a scientific theory, the contemporary ID is ready to (...) give up creationist explanation just to impeding Darwin’s fundamental idea according to which it’s possible to explain evolution only through a gradual material modification of structures. For comprehending a complex phenomenon as human language – according to ID – it’s necessary appealing to a second substance, whatever it is. This idea seems to be at the bottom of all those theories which have rejected monistic structural explanation for embracing functional, psychological or cultural accounts. We consider these kinds of explanation real unresolved residuals of ID, residuals nested in the heart of the most accredited scientific theories. (shrink)

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

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)

Minimal residual disease (MRD), a measure of residual cancer cells, is a concept increasingly employed in precision oncology, touted as a key predictive biomarker to guide treatment decisions. This paper critically analyzes the expanding role of MRD as a predictive biomarker in hematologic cancers. I outline the argument for MRD as a predictive biomarker, articulating its premises and the empirical conditions that must hold for them to be true. I show how these conditions, while met in paradigmatic cases (...) of MRD use in cancer, may not hold across other cancers where MRD is currently being applied, weakening the argument that MRD serves as an effective predictive biomarker across cancer medicine. (shrink)

This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids. An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has this closure property if its finitely subdirectly irreducible members form an elementary class. A syntactic characterization of this hypothesis is proved, and examples are discussed.

Thomas & Karmiloff- Smith show that the assumption of residual normality does not hold in connectionist simulations, and argue that RN has been inappropriately applied to childhood disorders. We agree. However, we suggest that the RN hypothesis may never have been fully viable, either empirically or computationally.

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)

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)

Progress in understanding the mind-body problem can be made without attempting to solve it as one unified problem, which it is not. Pepper's "Identity Theory" solution to the problem is now seen as not necessarily clarifying for the question of dualism. Residual asymmetrical dualism is proposed as a theory offering one very good way to think about this set of problems in a variety of modes of inquiry. These include neurophysiological research on the amygdala by LeDoux, research in the (...) phenonenon of hearing and learning while under general anesthetic, Gendlin's methods of focusing upon the body during therapeutic procedures and during creative composition of poetry, and Dewey's position concerning "primary experience" versus a "secondary pseudo-environment" inhabited by the civilized human. Residual asymmetrical dualism is not a value-neutral theory: it is based on a determination that bodily intelligence must ultimately guide mental functioning if survival and well-being are to be secured. It leads to take actions within society to carry out whatever steps are needed to alleviate the mind-body split whenever such a split is harmful to human interaction. (shrink)

Growth mixture models are regularly applied in the behavioral and social sciences to identify unknown heterogeneous subpopulations that follow distinct developmental trajectories. Marcoulides and Trinchera recently proposed a mixture modeling approach that examines the presence of multiple latent classes by algorithmically grouping or clustering individuals who follow the same estimated growth trajectory based on an evaluation of individual case residuals. The purpose of this article was to conduct a simulation study that examines the performance of this new approach for determining (...) the number of classes in growth mixture models. The performance of the approach to correctly identify the number of classes is examined under a variety of longitudinal data design conditions. The findings demonstrated that the new approach was a very dependable indicator of classes across all the design conditions considered. (shrink)

This work extend to residuated lattices the results of [ 7 ]. It also provides a possible generalization to this context of frontal operators in the sense of [ 9 ]. Let L be a residuated lattice, and f : L k → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L . We use this characterization of compatible functions in order to prove that the variety of (...) residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P ( x , y ) and Q ( x , y ) on a residuated lattice L which imply that the function $${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators. (shrink)

In this paper, we propose and axiomatically characterize residual contractions, a new kind of contraction operators for belief bases. We establish that the class of partial meet contractions is a strict subclass of the class of residual contractions. We identify an extra condition that may be added to the definition of residual contractions, which is such that the class of residual contractions that satisfy it coincides with the class of partial meet contractions. We investigate the interrelations (...) in the sense of inclusion among the class of residual contractions and other classes of well known contraction operators for belief bases. (shrink)

This paper introduces the concept of moral residue to global health, and shows how its presence undermines crucial interventions and research, especially in the global south. Lingering feelings of anxiety, anger, blame or frustration often exist among local populations, where previous interventions or research have left traces of harm and/or exploitation. The existence of such feelings reflects the presence of moral residue, recognizing the moral experiences of epistemic injustices, which in turn undermines critical interventions and research through outright rejection or (...) passive non-compliance among affected populations. While such situations have been variously interpreted and relevant strategies developed to address the issues, little to no consideration is made on the implications of moral residue experiences in global health contexts and how to address them. This paper demonstrates the presence of moral residue in global health and proffers an African ethical approach, a harmony framework, for addressing moral residue issues, as part of a holistic approach towards tackling population health crises without compromising health gains for affected populations in the global south. (shrink)

In response to our target article, many of the commentators concentrated on our notion of Residual Normality. In our response, we focus on the questions raised by this idea. However, we also examine broader issues concerning the importance of incorporating a realistic theory of the process of development into explanations of developmental deficits.

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 discusses Crawley completions of residuated lattices. While MacNeille completions have been studied recently in relation to logic, Crawley completions (i.e. complete ideal completions), which are another kind of regular completions, have not been discussed much in this relation while many important algebraic works on Crawley completions had been done until the end of the 70’s. In this paper, basic algebraic properties of ideal completions and Crawley completions of residuated lattices are studied first in their conncetion with the join (...) infinite distributivity and Heyting implication. Then some results on algebraic completeness and conservativity of Heyting implication in substructural predicate logics are obtained as their consequences. (shrink)

Thomas & Karmiloff- Smith ’ argument that the Residual Normality assumption is not valid for developmental disorders has implications for models of cognition in schizophrenia, a disorder that may involve a neurodevelopmental pathogenesis. A limiting factor for such theories is the lack of understanding about the nature of the cognitive system. Moreover, it is unclear how the proposal that modularization emerges from developmental processes would change that fundamental question.

Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which (...) is a copy of the {, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops). (shrink)

Two recent defences of the recovery postulate for contraction of belief sets are analyzed. It is concluded that recovery is defensible as a by-product of a formalization that is idealized in the sense of being simplified for the sake of clarity. However, recovery does not seem to be a required feature of the doxastic behaviour of ideal (perfectly rational) agents. It is reasonable to expect that there should be epistemic residues (remnants of rejected beliefs), but not that these should always (...) suffice to recover all the original beliefs if the discarded belief is reinstated. (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)

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.

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)

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

Given a pseudovariety [MATHEMATICAL SCRIPT CAPITAL C], it is proved that a residually-[MATHEMATICAL SCRIPT CAPITAL C] superstable group G has a finite seriesG0 ⊴ G1 ⊴ · · · ⊴ Gn = Gsuch that G0 is solvable and each factor Gi +1/Gi is in [MATHEMATICAL SCRIPT CAPITAL C] . In particular, a residually finite superstable group is solvable-by-finite, and if it is ω -stable, then it is nilpotent-by-finite. Given a finitely generated group G, we show that if G is ω (...) -stable and satisfies some residual properties , then G is finite. (shrink)

This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which (...) either treated fewer subsignatures or focussed on the conservation of theorems only. (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)

We present a unified vision of human knowledge, the external world and ourselves in the frame of an overall unity of Everything. Two main sources of knowledge are considered to this goal: an admittedly reductionist version of Modern Science and a few key elements of Oriental Philosophy. Our view is based on an analogy between the fundamental unity of Vedic ontology and the Grand Unification scheme of Particle Physics traced along the evolution of the Universe. Our key statement is that (...) these two sources of knowledge describe the same ontological story of separation, from an original One down to the multiplicity of the phenomenological world. We further substantiate this vision by drawing an analogy between the Residue of the Vedic sacrifice and the post-Big-Bang cosmological Matter-Antimatter Asymmetry. We also discuss a Big-Bang analogy between the postulated field of Inflationary Cosmology (the Inflaton) and the Vedic dissolution-decomposition of the cosmic progenitor Prajapati – whose body, according to Brahmanas texts, provides the material substrate of which the Universe is built. (shrink)

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.

Bad things happen; and not just to bad people, but with apparent indifference to the moral or other qualities of the victims. For the theist who believes that the world is created and governed by an all-powerful and perfectly good God, this is a notorious difficulty. In fact the problem of evil is surely the most persuasive consideration available in favour of atheism.

Transplant coordinators play a pivotal role in the process of obtaining consent for live or dead donation of organs. The objective of the project is to unveil emotional experiences and ethical conduct of transplant coordinators using a qualitative research methodology. Ten transplant coordinators who have worked for more than 20 years in this job were recruited by using a purposive sampling technique. The transplant coordinators spoke of negative feelings and moral distress with regard to futile care of family members of (...) deceased donors as well as of living donors. Transplant coordinators experience moral distress on a daily basis; being compelled to compromise their integrity causes moral distress and moral residue, hence, training and support should be offered to them. (shrink)

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) (...) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

A residuated lattice is said to be integrally closed if it satisfies the quasiequations \ and \, or equivalently, the equations \ and \. Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping \\backslash {\mathrm {e}}\) on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with respect (...) to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic. (shrink)

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the (...) extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. (shrink)