A theory of relations is presented that provides a detailed account of the logical structure of relational complexes. The theory draws a sharp distinction between relational complexes and relational states. A salient difference is that relational complexes belong to exactly one relation, whereas relational states may be shared by different relations. Relational complexes are conceived as structured perspectives on states ‘out there’ in reality. It is argued that only relational complexes have occurrences of (...) objects, and that different complexes of the same relation may correspond to the same state. (shrink)

The core issue of our target article concerns how relational complexity should be assessed. We propose that assessments must be based on actual cognitive processes used in performing each step of a task. Complexity comparisons are important for the orderly interpretation of research findings. The links between relational complexity theory and several other formulations, as well as its implications for neural functioning, connectionist models, the roles of knowledge, and individual and developmental differences, are considered.

Working memory limits are best defined in terms of the complexity of the relations that can be processed in parallel. Complexity is defined as the number of related dimensions or sources of variation. A unary relation has one argument and one source of variation; its argument can be instantiated in only one way at a time. A binary relation has two arguments, two sources of variation, and two instantiations, and so on. Dimensionality is related to the number of (...) chunks, because both attributes on dimensions and chunks are independent units of information of arbitrary size. Studies of working memory limits suggest that there is a soft limit corresponding to the parallel processing of one quaternary relation. More complex concepts are processed by or In segmentation, tasks are broken into components that do not exceed processing capacity and can be processed serially. In conceptual chunking, representations are to reduce their dimensionality and hence their processing load, but at the cost of making some relational information inaccessible. Neural net models of relational representations show that relations with more arguments have a higher computational cost that coincides with experimental findings on higher processing loads in humans. Relational complexity is related to processing load in reasoning and sentence comprehension and can distinguish between the capacities of higher species. The complexity of relations processed by children increases with age. Implications for neural net models and theories of cognition and cognitive development are discussed. (shrink)

This commentary focusses on the evidence used by Halford et al. to support their postulated links between relational complexity and age differences in children's understanding of concepts. None of their developmental claims is consistent with recent cognitive-developmental research. Relational complexity must be an important variable in cognition, but it does not provide a satisfactory metric for explaining cognitive development.

Halford et al.'s analysis of relational complexity provides a possible framework for characterizing the symbolic functions of the prefrontal cortex. Studies of prefrontal patients have revealed that their performance is selectively impaired on tasks that require integration of two binary relations (i.e., tasks that Halford et al.'s analysis would identify as three-dimensional). Analyses of relational complexity show promise of helping to understand the neural substrate of thinking.

Richard Miller uses the concepts of alterity and intimacy as touchstones for analyzing neglected aspects of our interpersonal and social relationships. He argues that, as persons in relation, we oscillate between experiences of alterity and intimacy, and it is with a greater awareness of this oscillation that we do best to consider our ethical responsibilities. This paper affirms the value of thinking about—and potentially reimagining—how we conceive and relate to various others. It also makes explicit that, as persons, each of (...) us is separate, not only from some, but from all other persons, even as we are also one with them. Moreover, each of us is different from all other persons, even as we are also like them. The aspects of persons and relationships on which we focus, in a given situation, matter because they partly determine the choices that we make in another’s regard. (shrink)

Relational complexity theory has important virtues, but the present model omits key aspects and evidence. In contrast, skill theory specifies (1) a detailed series of developmental changes in relational complexity from birth to age 30, (2) processes of interaction of content and structure that produce variability in complexity, (3) the role of cortical development, and (4) empirical criteria for complexity levels, including developmental discontinuities. Many findings support these specifications.

This paper introduces a new method of interpreting complex relation terms in a second-order quantified modal language. We develop a completely general second-order modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting n-place relations. Several issues arise in connection with previous, algebraic methods for interpreting the relation terms. The new method of interpreting these terms described here addresses those issues while establishing an interesting connection between λ and ε calculi. The resulting (...) semantics provides a precise understanding of the theory of relations. (shrink)

We discuss how modified dual-task approaches may be used to verify the degree to which cognitive tasks are capacity demanding. We also delineate some of the complexities associated with the use of the “double easy-to-hard” paradigm for testing claim of Halford, Wilson & Phillips that hierarchical reasoning imposes processing demands equivalent to those of transitive reasoning.

Mental state reasoning or theory-of-mind has been the subject of a rich body of imaging research. Although such investigations routinely tap a common set of regions, the precise function of each area remains a contentious matter. With the help of functional magnetic resonance imaging (fMRI), we sought to determine which areas are involved when processing mental state or intentional metarepresentations by focusing on the relational aspect of such representations. Using non-intentional relational representations such as spatial relations between persons (...) and between objects as a contrast, the results ascertained the involvement of the precuneus, the temporal poles, and the medial prefrontal cortex in the processing of intentional representations. In contrast, the anterior superior temporal sulcus and the left temporo-parietal junction were implicated when processing representations that refer to the presence of persons in relational contexts in general. The right temporo-parietal junction, however, was specifically activated for persons entering spatial relations. The level of representational complexity, a previously unexplored factor, was also found to modulate the neural response in some brain regions, such as the medial prefrontal cortex and the right temporo-parietal junction. These findings highlight the need to take into account the critical roles played by an extensive network of neural regions during mental state reasoning. (shrink)

This article reports a study carried out in order to measure how semantic factors affect reductions in the difficulty of the Chinese Ring Puzzle (CRP) that involves removing five objects according to a recursive rule. We hypothesised that semantics would guide inferences about action decision making. The study involved a comparison of problem solving for two semantic isomorphic variants of the CRP ( fish and fleas ) with problem solving for the puzzle's classic variant (the Balls and Boxes problem; Kotovsky (...) learning involves generalising the relations within a coherent, interconnected whole that makes up the puzzle's rules. It seems that semantics make it easier for the puzzle's elements to be grouped together within coherent, meaningful wholes, which reduces relational complexity and facilitates problem solving. (shrink)

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory \(\rm S_2 ^1(PV)\) has bounded model companion then \(\rm NP=coNP\). We also study bounded versions of some other related notions such as Stone topology.

An application of the Method of Analysis of Relational Complexity (MARC) to suppositional reasoning in the knight-knave task is outlined. The task requires testing suppositions derived from statements made by individuals who either always tell the truth or always lie. Relational complexity (RC) is defined as the number of unique entities that need to be processed in parallel to arrive at a solution. A selection of five ternary and five quaternary items were presented to 53 psychology (...) students using a pencil and paper format. A computer-administered version was presented to 50 students. As predicted, quaternary problems were associated with higher error rates and longer response times than ternary problems. The computer-administered form was more difficult than the pencil and paper version of the test. These differences are discussed in terms of RC theory and alternative processing accounts. Together, they indicate that the relational complexity metric is a useful and parsimonious way to quantify complexity of reasoning tasks. (shrink)

The fundamental relations that underlie cognitive comparisons—“same” and “different”—can be defined at multiple levels of abstraction, which vary in relational complexity. We compared response times to decide whether or not two sequentially‐presented patterns, each composed of two pairs of colored squares, were the same at three levels of abstraction: perceptual, relational, and system (higher order relations). For both 150 ms and 5 s inter‐stimulus intervals (ISIs), both with and without a masking stimulus, decision time increased with level (...) of abstraction. Sameness at lower complexity levels contributed to decisions based on the higher levels. The pattern of comparison times across levelswas not predictable solely from encoding times. The results indicated that relations at multiple levels of complexity can be abstracted and compared in working memory, with higher complexity levels requiring more processing time. We simulated the impact of relational complexity on response time using Learning and Inference with Schemas and Analogies (LISA), a computational model of relational comparisons based on dynamic binding of elements into roles in a relational working memory. (shrink)

Two assertions of Halford et al. are critiqued: their claim of priority in relational complexity analysis and the sufficiency for cognitive development of their relational-complexity analysis of tasks. Critical discussion of concrete task analyses (i.e., the relational complexity of proportionality problems, of balance scale problems, and the Tower of Hanoi) serves, by way of counterexamples, to highlight problems in their method.

The fundamental relations that underlie cognitive comparisons—“same” and “different”—can be defined at multiple levels of abstraction, which vary in relational complexity. We compared response times to decide whether or not two sequentially‐presented patterns, each composed of two pairs of colored squares, were the same at three levels of abstraction: perceptual, relational, and system (higher order relations). For both 150 ms and 5 s inter‐stimulus intervals (ISIs), both with and without a masking stimulus, decision time increased with level (...) of abstraction. Sameness at lower complexity levels contributed to decisions based on the higher levels. The pattern of comparison times across levelswas not predictable solely from encoding times. The results indicated that relations at multiple levels of complexity can be abstracted and compared in working memory, with higher complexity levels requiring more processing time. We simulated the impact of relational complexity on response time using Learning and Inference with Schemas and Analogies (LISA), a computational model of relational comparisons based on dynamic binding of elements into roles in a relational working memory. (shrink)

Measurements of the dimensionality of chaotic attractors obtained on behavioral data represent the task complexity and also could be hypothesized to reflect the number of synchronized neural groups involved in the generation of the data. The changes in dimensionality for different experimental conditions suggest that limited processing capacity, task complexity, demand, and synchrony in neural firing might be closely related.

The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider the class $$\mathsf{TPA}^-$$ (...) TPA - of the relation algebra reducts of $$\mathsf{TPA}$$ TPA ’s, as well. We prove that the equational theory of $$\mathsf{TPA}^-$$ TPA - is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

BackgroundRespect for autonomy is a paramount principle in end-of-life ethics. Nevertheless, empirical studies show that decision-making, exclusively focused on the individual exercise of autonomy fails to align well with patients’ preferences at the end of life. The need for a more contextualized approach that meets real-life complexities experienced in end-of-life practices has been repeatedly advocated. In this regard, the notion of ‘relational autonomy’ may be a suitable alternative approach. Relational autonomy has even been advanced as a foundational notion (...) of palliative care, shared decision-making, and advance-care planning. However, relational autonomy in end-of-life care is far from being clearly conceptualized or practically operationalized.Main bodyHere, we develop a relational account of autonomy in end-of-life care, one based on a dialogue between lived reality and conceptual thinking. We first show that the complexities of autonomy as experienced by patients and caregivers in end-of-life practices are inadequately acknowledged. Second, we critically reflect on how engaging a notion of relational autonomy can be an adequate answer to addressing these complexities. Our proposal brings into dialogue different ethical perspectives and incorporates multidimensional, socially embedded, scalar, and temporal aspects of relational theories of autonomy. We start our reflection with a case in end-of-life care, which we use as an illustration throughout our analysis.ConclusionThis article develops a relational account of autonomy, which responds to major shortcomings uncovered in the mainstream interpretation of this principle and which can be applied to end-of-life care practices. (shrink)

I relate plural quantification, and predicate logic where predicates do not need a fixed number of argument places, to the part-whole relation. For more on these themes see later work by Boolos, Lewis, and Oliver & Smiley.

Animal Suffering and Public Relations conducts an ethical assessment of public relations, mainly persuasive communication and lobbying, as deployed by some of the main businesses involved in the animal industrial complex - the industries participating in the systematic and institutionalized exploitation of animals. Society has been experiencing a growing ethical concern regarding humans' (ab)use of other animals. This is a trend first promoted by the development of animal ethics - which claims any sentient being, because of sentience, deserves moral consideration (...) - and more recently by other approaches from the social sciences, including critical animal studies. In this volume, we aim to start an entirely unaddressed discussion within the field of public relations: the need to problematize the ethics of persuasion when nonhuman animal suffering is involved, particularly the impact of persuasion and lobbying on compassion towards other animals in the cases of food, experimentation, entertainment and environment management. The books provides an interdisciplinary, theoretical discussion illustrated with international case studies from experts in strategic communication, public relations, lobbying and advocacy, animal ethics, philosophy of law, political philosophy and social psychology. This unique book merges the fields of critical public relations, animal ethics and critical animal studies and will be of direct appeal to a wide range of researchers, academics and doctoral students across related fields. (shrink)

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and (...) from both below and above. Then we show that [Formula: see text] is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

In this paper I present a formal language in which complex predicates stand for properties and relations, and assignments of denotations to complex predicates and assignments of extensions to the properties and relations they denote are both homomorphisms. This system affords a fresh perspective on several important philosophical topics, highlighting the algebraic features of properties and clarifying the sense in which properties can be represented by their extensions. It also suggests a natural modification of current logics of properties, one in (...) which some complex predicates stand for properties while others do not. (shrink)

Color -order systems highlight certain features of color phenomenology while neglecting others. It is misleading to speak as if there were a single “psychological color space” that might be described by a rather simple formal structure. Criticisms of functionalism based on multiple realizations of a too-simple formal description of chromatic pheno-menal relations thus miss the mark. It is quite implausible that a functional system representing the full complexity of human color phenomenology should be realizable by radically different qualitative states.

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔fS f].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial (...) timem-reducibility on exponential time sets, which is${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is${\rm{\Sigma }}_4^0$. (shrink)

A central feature of complexity is that it is based on non-linear, recursive relations. However, in most current accounts of complexity such relations, while non-linear, are based on the reductive relations of a Newtonian onto-epistemological framework. This means that the systems that are emergent from the workings of such relations are a narrowly reduced spectrum of complex systems. It is argued that John Dewey’s trans-actional relations, relations that are characterized by an irreducible internal distinction, can function as an (...) exemplar of the less reductive relations seen in living complex systems. Conceptualizing complex relations and complex systems in a less reductive way allows indeterminate concepts such as epistemological limits and the generative production of meaning, which are inextricably linked, to be considered. Methodologically, such lessened reduction highlights functional systems, revealing aspects of the biopsychosocial human ‘complex system of complex systems’ that are hidden from view in a Newtonian framework. (shrink)

Analogy is an important cognitive process that has been researched extensively. Functional accounts of it typically involve at least four stages of processing ; however, these accounts take the way in which the base analogue is understood, along with its relational structure, for granted. The goal of this paper is to open up a discussion about how this process may occur. To this end, this paper describes two experiments that vary the level of relational complexity across exemplars. (...) It was found that relational recognition tasks benefit from increased complexity, while mapping tasks suffer from it. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the (...) permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures. (shrink)

This paper examines and classifies the computational complexity of model checking and satisfiability for hybrid logics over frames with equivalence relations. The considered languages contain all possible combinations of the downarrow binder, the existential binder, the satisfaction operator, and the global modality, ranging from the minimal hybrid language to very expressive languages. For model checking, we separate polynomial-time solvable from PSPACE-complete cases, and for satisfiability, we exhibit cases complete for NP, PS pace , NE xp T ime , and (...) even N2E xp T ime . Our analysis includes the versions of all these languages without atomic propositions, and also complete frames. (shrink)

The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) as well. We prove that the equational theory of \ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

Let$ \le _c $be computable the reducibility on computably enumerable equivalence relations. We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$,$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$ is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set of (...) the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation. (shrink)

We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity, and interpolation problems over the intuitionistic logic Int (...) and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras. In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int + . In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi. (shrink)

The structural and dynamic dimensions of complexity of the Internet are connected with epistemological and ontological factors, which are the main modes of complexity of the sciences of the Internet. These dimensions and modes of complexity are relevant for the communication sciences, because this field is one of the most important areas of development of this network of networks. Philosophy needs to address the problems of complexity that arise from the sciences of the Internet that have (...) consequences for the communication sciences. This article deals with these problems through a new approach in the philosophy of communication sciences based on the sciences of the artificial conceived as sciences of design. (shrink)

We study partitions of Fraïssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin.

This article considers the question “Is There, Can There Be, such an Activity as World Humanities?” My response takes issue with the phrase “world humanity” that serves to legitimize a homogenizing process of globalization while simultaneously hiding from view the injustices and power imbalances that globalization creates. From my perspective as a Cree-Métis scholar, the world humanities are only valuable if they participate in the recuperation and protection of Indigenous stories—and by extension Indigenous languages and epistemes—that would require a reconsideration (...) of the definition and obligations of what it means to be human. (shrink)

The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations R and S on natural numbers, R is computably reducible to S if there is a computable function f:ω→ω that induces an injective map from R-equivalence classes to S-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the (...) polynomial-time reducibility. In this work, we explore Peq, the degree structure generated by primitive recursive reducibility on primitive recursive equivalence relations with infinitely many equivalence classes. In contrast with all other known degree structures on equivalence relations, we show that Peq has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of Peq, proving, e.g., that the structure is neither rigid nor homogeneous. (shrink)