Juulia Lahdenperä
email: juulia.lahdenpera@helsinki.fi

Comparing male and female students' self-efficacy and self-regulation skills in two undergraduate mathematics course contexts

Keywords: teachers' and students' practices at university level, novel approaches to teaching, self-efficacy beliefs, self-regulation, instructional design

Students' efficacy beliefs have a positive influence on students' academic achievement and retention, especially for female students. These beliefs are closely linked to students' ability to regulate their learning. In this quantitative study, students' self-efficacy and self-regulation of learning are compared in two university mathematics courses that differ in content but also in their pedagogical setting: one is a more traditional lecture-based course, and the other course is taught with the Extreme Apprenticeship (XA) method. The analysis is based on the same cohort of students in the two contexts (N=91). The results suggest that students have higher self-efficacy levels in the course using the XA method. Also, the XA course seems to diminish some gender differences present in the more traditional course setting.

INTRODUCTION

According to [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] view of situated learning, a set of skills as well as a set of values and perspectives are needed for a holistic understanding of a topic. [START_REF] Collins | Cognitive apprenticeship: Making thinking visible[END_REF] suggest that the process of acquiring this kind of understanding occurs most naturally within the community possessing this knowledge. In this light, it is not irrelevant what kind of instructional practices are implemented to teach university mathematics, and how do these practices offer students opportunities to participate. As Greeno (1997, p. 9) states, "methods of instruction are not only instruments for acquiring skills; they also are practices in which students learn to participate". Therefore, to enhance students' learning, instructional designs used to teach university mathematics should offer students both an opportunity to acquire knowledge and an opportunity to become part of a community.

Partly to answer to this need, lots of effort has been put into developing the educational setting at the Department of Mathematics and Statistics at the University of Helsinki (see eg. [START_REF] Oikkonen | Ideas and results in teaching beginning maths students[END_REF][START_REF] Rämö | Extreme Apprenticeship -Emphasising Conceptual Understanding in Undergraduate Mathematics[END_REF]. The department's teaching has gone through major changes during the past few years as many of the undergraduate level courses are now using the Extreme Apprenticeship (XA) method as their pedagogical framework. In addition, some of the courses working within the traditional lecture-based framework have been developed towards a more interactive direction. This paper approaches the development of university mathematics education from the perspective of efficacy beliefs and self-regulation of learning. As a part of a larger research project aiming at comparing university mathematics teaching practices and transferring knowledge from research into practice, this paper elaborates on students' experiences of different instructional designs with the focus on their efficacy beliefs and self-regulation of learning.

THEORETICAL FRAMEWORK

Self-efficacy is a person's belief about how well they can perform a specific task in a specific context; these beliefs determine how people feel, think, motivate themselves and behave [START_REF] Bandura | Self-efficacy[END_REF]. Self-efficacy beliefs play a crucial role in learning mathematics as self-efficacy enhances academic achievement [START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF][START_REF] Peters | Examining the relationships among classroom climate, selfefficacy, and achievement in undergraduate mathematics: a multilevel analysis[END_REF], especially in female students [START_REF] Raelin | The gendered effect of cooperative education, contextual support, and self-efficacy on undergraduate retention[END_REF]. The gender aspect of selfefficacy is relevant as it affects female students' career choices and increases their retention in STEM fields [START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF][START_REF] Raelin | The gendered effect of cooperative education, contextual support, and self-efficacy on undergraduate retention[END_REF]. In terms of instructional design, [START_REF] Peters | Examining the relationships among classroom climate, selfefficacy, and achievement in undergraduate mathematics: a multilevel analysis[END_REF] shows that students' self-efficacy is higher in teacher-centred than in learner-centred classroom. However, [START_REF] Kogan | Assessing long-term effects if inquiry-based learning: a case study from college mathematics[END_REF] argue that female students obtain affective gain from student-centred courses as they are more confident in their mathematical abilities in these kinds of contexts.

The notion of self-regulation characterises how students regulate their cognition, behaviour, motivation and emotions to enhance their personal learning processes [START_REF] Pintrich | A conceptual framework for assessing motivation and selfregulated learning in college students[END_REF]. Students are expected to learn self-regulation skills during their university studies and therefore instructional designs should support the development of these skills [START_REF] Coertjens | Modeling change in learning stratgies throughout higher education: A multi-indicator latent growth perspective[END_REF]. The self-regulation process is cyclical in nature as feedback from prior performance is used to adjust future learning performances [START_REF] Pintrich | A conceptual framework for assessing motivation and selfregulated learning in college students[END_REF][START_REF] Zimmerman | Attaining self-regulation: A social cognitive perspective[END_REF]. Consequently, the quality of selfregulated learning is supported by motivational beliefs, such as self-efficacy beliefs [START_REF] Heikkilä | Studying in higher education: students' approaches to learning, self-regulation, and cognitive strategies[END_REF][START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF]. As the social aspect of learning has a significant role in learning self-regulation skills [START_REF] Volet | Self-and social regulation in learning contexts: An integrative perspective[END_REF], instructional designs should also encourage student collaboration.

Aims and research questions

The aim of the paper is to compare students' self-efficacy and self-regulation of learning in two different course contexts. The further analysis focuses on gender as previous research shows that especially female students benefit from more studentcentred course designs. The research questions are:

1. How do self-efficacy and self-regulation of learning differ between the two course settings?

2. How is the course setting related to male and female students' self-efficacy and self-regulation of learning?

METHODOLOGY

This study approaches the research questions with a quantitative analysis of students' self-efficacy and self-regulation of learning in two course contexts. The following subsections move on to describe the context, the data collection procedure and data analysis in greater detail.

Context

The research was conducted in a research-intensive university in Finland. Data was collected from two different courses that students usually take during the first semester of their university mathematics studies. Both courses are proof-based, sixweek and five-credit (ECTS) courses with over 200 students. In addition to mathematics majors, the courses are taken by many students studying mathematics as their minor subject; these students are usually majoring in physics, computer science, chemistry or education. The two courses, course A and course XA, are implemented in accordance with different pedagogical frameworks. The main difference in the course implementations are the role of lectures, design of the tasks, and the form of support given to the students by the teaching assistants.

Course A is an analysis course. The main content of the course includes limit of a function, continuity, derivative, and its applications. It is necessary to point out that the course is an analysis course rather than a calculus course as exact definitions and proof construction are emphasised. The course functions within the traditional lecture-based setting. However, it has been developed over a decade towards a more interactive direction to respond to students' challenges in the beginning of their university mathematics studies.

The course A consists of four hours of lectures and four hours of small group sessions per week. The lectures are focusing on the main content of the course and aim at creating deep understanding behind those concepts. Inspired by Tall's three worlds of mathematics (see e.g. [START_REF] Tall | How humans learn to think mathematically: exploring the three worlds of mathematics[END_REF], the lectures are an active interplay between the human and formal sides of mathematics. The small group sessions are led by a teaching assistant, who is usually an older mathematics student. There are two different kinds of small group sessions. The other one is allocated to the problems students have solved prior to the class. The other small group session is allocated to solving a new set of problems during the session together with other students and with the help of the teaching assistant.

Course XA is a linear algebra and matrices course. The main content of the course includes general vector spaces, subspaces, linear mappings, and scalar products. In addition to mathematical content, the course emphasises skills such as reading mathematical text, oral and written communication, and proof construction. The course is taught with the XA method. The XA method is a student-centred educational method developed in the Department of Mathematics and Statistics and the Department of Computer Science at the University of Helsinki. The method emphasises learning by doing, personalised scaffolding and continuous feedback, and the core idea is to support students in becoming experts in their field by having them participate in activities that resemble those carried out by professionals (see eg. [START_REF] Rämö | Extreme Apprenticeship -Emphasising Conceptual Understanding in Undergraduate Mathematics[END_REF]. The XA method is constructed upon the ancient process of apprenticeship, where a skilled master supervises a novice apprentice, and its theoretical background is in situated view on learning and Cognitive Apprenticeship [START_REF] Collins | Cognitive apprenticeship: Making thinking visible[END_REF][START_REF] Rämö | Extreme Apprenticeship -Emphasising Conceptual Understanding in Undergraduate Mathematics[END_REF].

In the XA method, students learn skills and gain knowledge by working on tasks that have been divided into smaller and approachable goals, which are then merged together as the students start to master a topic. The main method of teaching is instructional scaffolding, and it is accompanied with continuous, bi-directional feedback. Further, it supports students to establish relations within the communities of practice which enhances the students' integration into the community [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF].

In practice, the teaching of the course consists of weekly problems, course material, guidance and three hours of lectures per week. There is a flipped learning approach as students start studying a new topic by solving a set of problems. These topics have not yet been discussed during the lectures, so students need to read the course material to complete the tasks. However, the tasks are designed to be approachable and there are teaching assistants specifically to this course helping the students in solving the problems. The teaching assistants guide the students in a learning space in the middle of the department in drop-in basis approximately six hours a day. Student collaboration is encouraged in the learning space. Students return written solutions to the problems every week. Few problems are selected for inspection and students get feedback for their solutions. The feedback focuses on solutions' logical structure, but also readability and language are evaluated, and students' have the possibility to improve and resubmit their solutions. Students are prepared when they come to lectures as they have done pre-lecture tasks. Lectures focus on active interaction as various small group activities are implemented and students' active participation encouraged. The aim is to form links between the topics and enhance holistic understanding. After the lectures students get more challenging problems on the topic.

Data collection

Quantitative data was collected on a five-point Likert scale (1=completely disagree, 5=completely agree) from students attending both courses. The questionnaire included items measuring students' approaches to learning, their experiences of the teaching-learning environment, self-efficacy and self-regulation of learning. In this paper, the analysis includes items measuring self-efficacy and self-regulation of learning from students who answered the questionnaire for both course contexts (N=91).

There are five items measuring self-efficacy. The items are slightly modified from [START_REF] Pintrich | A manual for the use of the Motivated Strategies for Learning Questionnaire[END_REF] and they are validated and highly used across disciplines in the Finnish context (see e.g. Parpala & Lindblom-Ylänne, 2012). The 15 items measuring self-regulation of learning are originally from the Inventory of Learning Styles (ILS, [START_REF] Vermunt | Inventory of Learning Styles in higher education: Scoring key[END_REF] and they have been modified to Finnish context [START_REF] Heikkilä | Studying in higher education: students' approaches to learning, self-regulation, and cognitive strategies[END_REF]. There self-regulation of learning is measured in four scales: self-regulation of process, self-regulation of content, external regulation, and lack of regulation. Self-regulation of process refers to a student's ability to regulate their own learning when facing challenges. Self-regulation of content measures student's seeking of additional literature beyond the course material. External regulation measures to what extent the lecturer regulates student's learning. Lack of regulation refers to possible problems in regulation of learning, such as not knowing how to proceed in the learning process or having challenges in finding ways to cover the course content.

Data analysis

The data analysis is conducted by using IBM Statistics 24. The data in this paper is a part of a larger data set. The results reported here are from the factor analyses computed for the larger data set.

At first, exploratory factor analysis (EFA) was conducted with principal axis factoring and a direct oblimin rotation. Based on the exploratory factor analysis, there are four factors measuring self-regulation of learning. This is in accordance with previous research (see eg. [START_REF] Heikkilä | Studying in higher education: students' approaches to learning, self-regulation, and cognitive strategies[END_REF]. Similarly, the factor structure of the self-efficacy scale is like in previous studies (see e.g. Parpala, & Lindblom-Ylänne, 2012) forming one factor. Boundaries used for Kaiser-Meyer-Olkin measure of sampling adequacy (KMO) was 0.7, and for Bartlett's test of sphericity p<0.001. One item measuring self-regulation of content was excluded from the factor based on a low communality, a mixed factor loading and deviant skewness and kurtosis. Every factor was then checked for internal consistency: the Cronbach's Alpha is 0.905 for the self-efficacy factor, 0.681 for the self-regulation of process factor, 0.671 for the self-regulation of content factor, 0.708 for the external regulation factor, and 0.661 for the lack of regulation factor. The reliabilities are above the 0.65 level which can be considered acceptable.

As the current study follows a repeated measures design, the data was analysed by using two-tailed paired samples t-test and Cohen's effect size d. In addition, one-way MANOVA with Wilk's Lambda was used to analyse the interaction of independent variables (gender) on the dependent variables (different course settings).

RESULTS

The data consists of 91 students (46 male, 45 female). There are also smaller mean differences in the self-regulation of process and selfregulation of content factors between the two course contexts (MD=0.14 and MD=-0.19 respectively). These differences are statistically significant on a 0.05 level (t(90)=2.189, p<0.05 and t(90)=-2.383, p<0.05 respectively). The results indicate that on average, students in course XA seek more actively additional literature beyond the course material and do more work than expected when compared to course A (self-regulation of content). In addition, an average student in course XA reports that they are more capable of regulating their learning processes when facing challenges when compared to course A. However, one must notice that the effect sizes are below 0.2 suggesting an insignificant role of the course contexts.

There is no statistically significant difference in the external regulation factor (MD= 0.06, t(90)=0.941, p=0.35). This means that on average, students report that the lecturers' instruction on how and in what order to proceed in learning the content influences their learning similarly in both course contexts.

Course Let's now move on to analyse both male and female students in one course context at a time. In course A context, male and female students differ statistically significantly in the self-efficacy factor (MD=-0.54, p<0.01, F(1,89)=6.602, partial η 2 =0.079). This difference is not present in course XA context (MD=0.04, p=0.821, F(1,89)=0.038, partial η 2 =0.001). These results indicate that female students have statistically significantly lower self-efficacy in course A compared to male students; however, this difference between genders is not present in course XA context.

There are statistically significant differences also in the external regulation factor. In course A context, the mean difference between male and female students is not statistically significant (MD=0.31, p=0.054, F(1,89)=3.827, partial η 2 =0.041). However, the p-value is very close to the 0.05-significance level. In course XA context, male and female students differ statistically significantly in the external regulation factor (MD=0.51, p<0.001, F(1,89)=12.947, partial η 2 =0.127). This means that, in course XA contexts, an average female student applies more external regulation compared to an average male student. In other words, female students report that the lecturers' instructions influence their learning more when compared to male students.

There are no statistically significant differences between male and female students in the two course contexts in self-regulation of process, self-regulation of content, and lack of regulation factors.

DISCUSSION

There are statistically significant differences between course A and course XA in relation to the self-efficacy and self-regulation scales. The biggest differences are in the self-efficacy and lack of regulation factors. The results show that an average student has higher self-efficacy levels in course XA than in course A, and that an average student lacks regulatory skills more often in course A than in course XA. This is supported by the effect sizes implying a moderate role for the course contexts when measuring self-efficacy and lack of regulation. The results bear significance as self-efficacy has a strong positive and lack of regulation a strong negative relation to academic performance [START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF][START_REF] Peters | Examining the relationships among classroom climate, selfefficacy, and achievement in undergraduate mathematics: a multilevel analysis[END_REF][START_REF] Vermunt | Relations between student learning patterns and personal and contextual factors and academic performance[END_REF].

Gender has a statistically significant interaction with the factor measuring selfefficacy. On average, female students report lower self-efficacy levels in course A compared to male students. In contrast, the self-efficacy levels are very similar for male and female students in course XA. In practice, an average female student is less confident in her abilities to succeed in the course A context when compared to an average male student. However, it seems that the change in course context diminishes the difference as there is no statistically significant difference present between genders in the self-efficacy factor in course XA context.

Female students report more external regulation compared to male students in both course contexts. The results are statistically significant only in course XA context, but the p-value is very close to the 0.05-significance level also in course A context. This means that the lecturers instructions have more influence on female students' learning processes than on male students' learning processes. This is supported by prior research [START_REF] Vermunt | Relations between student learning patterns and personal and contextual factors and academic performance[END_REF], although the current study does give any explanations to this phenomenon. However, in [START_REF] Vermunt | Relations between student learning patterns and personal and contextual factors and academic performance[END_REF] study there was no consistent interaction between students' gender and their learning patterns, and external regulation did not relate negatively to academic achievement. Further research is needed to understand the motivations behind this phenomenon, as well as its implications to instructional practices.

One of the major limitations of this study is that the two courses differ in content. The limitation is caused by the choice of research design as it was not possible to attain the same cohort of students in two different pedagogical settings with the same course content. However, the different course contents do not fully provide an explanation for the result that male and female students' ability to regulate and reflect on their learning processes is dependent on the course context. One can argue that the gender differences are not caused by the characteristics of the mathematics studied but by the characteristics of the learning environment used to study the mathematics. The results of this study may also be affected by the fact that some students are more capable to adopt themselves into new instructional designs. As argued by [START_REF] Kogan | Assessing long-term effects if inquiry-based learning: a case study from college mathematics[END_REF], student-centred course settings often feature collaborative work, problem-solving and communication, aspects known to be effective for female students. Also, [START_REF] Vermunt | Relations between student learning patterns and personal and contextual factors and academic performance[END_REF] states that female students like cooperative learning more compared to male students. In addition, students who have high confidence in collaboration seek more likely help from other students; these help-seeking students then perform better in a flipped mathematics classroom compared to students seeking less help [START_REF] Sun | The role of self-regulated learning in students' success in flipped undergraduate math courses[END_REF].

Despite of the limitations, the findings of this study are supported by prior research.

To conclude, the results propose that student-centred course designs support students' self-efficacy and self-regulation, especially in female students. More thorough analysis should be completed to understand the mechanisms and motivations behind the differences in these two course contexts and to draw more general conclusions regarding instructional designs used in teaching university mathematics.

  These students attended both course A and course XA. Students' scores on self-efficacy and self-regulation scales in both course contexts are presented in Table1with means, standard deviations, mean differences, paired-samples t-test for statistical significance, and Cohen's d for effect size.The biggest differences between the two course contexts lie in the self-efficacy and lack of regulation factors. Students report statistically significantly higher selfefficacy levels in course XA compared to course A (MD=0.58, t(90)=6.226, p<0.001). This means that students are more confident in their abilities to succeed in course XA compared to course A. The effect size (Cohen's d=0.62) implies a moderate role for the course context when measuring self-efficacy. A similar phenomenon occurs in the lack of regulation factor; students report statistically significantly less lack of regulation in course XA compared to course A (MD=0.48, t(90)=6.987, p<0.001). In practice this means that on average, students report that they lack regulation of learning more often in course A compared to course XA. In other words, it was easier for students to find ways to handle large quantities of content, self-evaluate their learning, and to meet the learning goals in course XA compared to course A. The effect size (Cohen's d=0.63) implies a moderate role for the course context when measuring lack of regulation.

Table 1 : Students' scores, mean differences and effect sizes on the self-efficacy and self- regulation factors in courses A and XA, as determined by two-tailed paired samples t- test (* for p<0.05 and *** for p<0.001 significance levels) and Cohen's d.

 1

		A Course XA
	Variable	Mean SD Mean SD	Mean difference	Effect
			(XA-A)	size
	Self-efficacy	3.09 0.96 3.67 0.86	0.58***	0.62
	Self-regulation of	2.83 0.82 2.97 0.80	0.14*	0.17

Table 2 : Differences between courses A and XA based on students' gender, as determined by one-way MANOVA (* for p<0.05, ** for p<0.01, and *** for p<0.001 significance levels).

 2

			Male	Female
	Factor	Course Mean SD Mean SD Mean difference
	Self-efficacy	A	3.36 0.96 2.82 0.89	-0.54**
		XA	3.65 0.96 3.69 0.72	0.04
	External regulation	A	3.43 0.80 3.74 0.74	0.31(*)
		XA	3.40 0.72 3.91 0.61	0.51***


Friedrich Wehrung
email: friedrich.wehrung01@unicaen.fr

RANGES OF FUNCTORS IN ALGEBRA

Keywords: August 10, 2018. 2000 Mathematics Subject Classification. 18A25, 18A20, 18A35, 03E05, 05D10, 06A07, 06A12, 06F20, 06F25, 08B10, 08A30 Category, functor, diagram, condensate, spectrum, sober space, spectral space, Stone duality, congruence, lattice, distributive, Murray -von Neumann equivalence, nonstable K-theory

Introduction

The present short survey paper deals with the following kind of problem. We are given classes A and B, together with a map Ψ : A → B. We would like to find a way to recognize the members of the range Ψ

[A] = def {Ψ(A) | A ∈ A} of Ψ. That is,
we would like to find simple criteria for a given B ∈ B to be Ψ(A) for some A ∈ A.

Our main line of argument is that in many situations, if there exists a solution A, then there exists a "nice" solution A nice , where "nice" means that part of the structure of B can be read on the construction of A nice .

In most of those situations, A and B are classes of structures: groups, rings, modules, lattices (which are all particular instances of so-called universal algebras), but also topological spaces. In all those instances, both A and B afford a notion of isomorphism, which enables us to state that two structures, although not identical in the set-theoretical sense, share all the properties that matter (e.g., isomorphic groups, homeomorphic topological spaces).

The relevant context, for this kind of problem, is thus category theory. Now A and B are both categories and Ψ is a functor from A to B. Moreover, instead of asking for Ψ(A) = B, we only ask for Ψ(A) ∼ = B (say that A lifts B with respect to Ψ), and the "range" of Ψ is the class of all B that are isomorphic to Ψ(A) for some object A of A: in formula,

Ψ[A] = def {B ∈ B | (∃A ∈ A)(Ψ(A) ∼ = B)} .
(1.1)

Observe right away that (1.1) already involves a quite common abuse of notation, used there for simplicity's sake: "B ∈ B" should be "B is an object of B", and, similarly, "A ∈ A" should be "A is an object of A". Calling the right hand side of (1.1) the "range" of Ψ is a similar abuse of terminology, as the Ψ[A] of (1.1) usually properly contains the set-theoretical range of Ψ.

"Niceness" of the solution A means that if B is obtained as a colimit of (usually simpler) structures B i , then A is also the colimit of suitably chosen structures A i with each Ψ(A i ) ∼ = B i . Thinking of the B i as all the "finitely generated" substructures of B, we would like to lift (with respect to Ψ) not only the object B, but the diagram formed by all the B i and the inclusion maps between them. We are thus trying to lift the slice category B ↓ B of all arrows from some object of B to B. Now it is often the case that we are only trying to lift a subdiagram of B ↓ B. A "diagram" in B is, really, a functor Φ : I → B, for some category I. Our initial problem can thus be recast as follows: we are given categories I, A, B together with functors Φ : I → B and Ψ : A → B. We are asking whether there exists a functor Γ : I → A such that the functors Φ and Ψ • Γ are isomorphic, in notation Φ ∼ = Ψ • Γ (two functors are isomorphic if there exists a natural transformation from one to the other all of whose components are isomorphisms; the relation Φ ∼ = Ψ • Γ could be paraphrased as "Φ = Ψ • Γ up to isomorphism").

The monograph Gillibert and Wehrung [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF] deals extensively with this kind of problem, in situations arising from algebra. It is articulated around a technical statement called there the Condensate Lifting Lemma (CLL for short, Lemma 3.4.2 in [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF]). This statement makes it possible, in many situations, to reduce the liftability of a diagram B of B to the liftability of a suitable object B of B, called a condensate of B. Even for diagrams indexed by a finite partially ordered set (poset for short) P , the construction of a condensate can be a difficult matter, relying on objects called lifters of P and whose existence follows from statements of infinite combinatorics based on Kuratowski's Free Set Theorem (cf. Kuratowski [START_REF] Kuratowski | Sur une caractérisation des alephs[END_REF], Erdős et al. [START_REF] Erdős | Combinatorial Set Theory: Partition Relations for Cardinals[END_REF]Theorem 46.1]). The "size" (often defined as cardinality) of a condensate may be (strictly) larger than the size of the diagram B: even in case B is a diagram of finite structures, indexed by a finite poset P , the condensate B may have transfinite cardinality, often ℵ n-1 where n is nothing else than the order-dimension of the poset P (cf. Gillibert and Wehrung [START_REF]An infinite combinatorial statement with a poset parameter[END_REF]). In addition, this construction works if P is a finite lattice, but may fail for general finite posets.

Many readers would object that this is sounding a bit abstract. Hence, let us now inch away from category theory and present some more specific situations.

Spectral spaces: Stone duality and Hochster's Theorem

For more details and references about the present section, we refer the reader to Grätzer [15, § 2.5], Hochster [START_REF] Hochster | Prime ideal structure in commutative rings[END_REF], Johnstone [18, § II.3 and Ch. V], Stone [START_REF] Marshall | Topological representations of distributive lattices and Brouwerian logics[END_REF].

2.1.

Hochster's Theorem. The Zariski spectrum of a commutative unital ring A, is a topological space, fundamental in algebraic geometry. Let us recall the definition of that space. An ideal P of A is prime if it is proper (i.e., P = A, equivalently 1 / ∈ P ) and the quotient ring A/P is a domain (equivalently, xy ∈ P implies that either x ∈ P or y ∈ P , whenever x, y ∈ A). We denote by Spec A the set of all prime ideals of A, endowed with the topology whose closed sets are exactly those of the form Spec(A, X)

= def {P ∈ Spec A | X ⊆ P } , (2.1)
for X ⊆ A. This is a so-called hull-kernel topology.

The assignment A → Spec A can be extended to a (contravariant) functor, by sending any homomorphism f : A → B of commutative unital rings to the map Spec

f : Spec B → Spec A, Q → f -1 [Q] = def {x ∈ A | f (x) ∈ Q}.
This map is easily seen to be continuous (this is not the best possible guess, but it will do for now). Hence, Spec defines a contravariant functor, from the category of all commutative unital rings, with unital ring homomorphisms, to the category of all topological spaces, with continuous maps.

What kind of topological space is Spec A? Easy examples show that it may not be Hausdorff. However, other easy examples show that not every topological space is the Zariski spectrum of a commutative unital ring, even in the finite case. Let us now prepare for the definition of the relevant topological spaces.

Definition 2.1. A nonempty closed set F in a topological space X is irreducible if F = A ∪ B implies that either F = A or F = B,
for all closed sets A and B. We say that X is sober if every irreducible closed set is the closure of a unique singleton 1 . Definition 2.1 can be paraphrased by saying that a topological space is sober if it has as few points as possible with respect to the lattice structure of its collection of open sets.

For a topological space X, we shall denote by • K(X) the set of all open and compact 2 subsets of X, partially ordered under set inclusion. In general, for arbitrary

U, V ∈ • K(X), the union U ∪ V always belongs to • K(X). However, the intersection U ∩ V may not belong to • K(X). Definition 2.2. A topological space X is spectral if it is sober and • K(X)
is a basis of the topology of X, closed under finite intersection.

Defining the empty intersection as the whole space X, this shows that a spectral space is always compact.

It is well know, and easy to verify, that the Zariski spectrum of any commutative unital ring is a spectral space. The converse is given by the following theorem of Hochster [START_REF] Hochster | Prime ideal structure in commutative rings[END_REF].

Theorem 2.3 (Hochster). Every spectral space is homeomorphic to the Zariski spectrum of some commutative unital ring.

Hochster's construction, that assigns, to any spectral topological space X, a commutative unital ring A X such that X ∼ = Spec A X , is functorial, from the category of all commutative unital rings with unital ring embeddings, to spectral spaces with spectral maps. By definition, for spectral spaces X and Y , a map ϕ :

X → Y is spectral if ϕ -1 [V ] is compact open whenever V is a compact open subset of Y ;
every spectral map is continuous, but not every continuous map is spectral.

In particular, Hochster's Theorem implies that the range of the Zariski spectrum functor is completely described: it is the class of all spectral topological spaces.

Stone duality.

The construction of the Zariski spectrum, for commutative unital rings, can be extended to bounded distributive lattices, in the following way. Recall that a lattice is a structure (L, ∨, ∧), where ∨ and ∧ are both binary operations on a set L such that there is a partial ordering ≤ for which x ∨ y = sup(x, y) (the join of {x, y}) and x ∧ y = inf(x, y) (the meet of {x, y}) whenever 1 Due to the uniqueness, every sober space is T 0 (not all references assume this). 2 Throughout the paper, "compact" means what some other references call "quasicompact" (i.e., every open cover has a finite subcover); in particular, it does not imply Hausdorff.

x, y ∈ L. Necessarily, x ≤ y ⇔ x ∨ y = y ⇔ x ∧ y = x whenever x, y ∈ L, so the partial ordering ≤ is uniquely determined by either ∨ or ∧. We say that L is

• distributive if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) whenever x, y, z ∈ L;
• bounded if ≤ has a smallest element (then denoted by 0) and a largest element (then denoted by 1). An ideal P , in a bounded distributive lattice D, is prime if it is proper (i.e., P = D, equivalently 1 / ∈ P ) and x ∧ y ∈ P implies that either x ∈ P or y ∈ P , whenever x, y ∈ D.

The remainder of the definition of the spectrum, of a bounded distributive lattice, follows the one for commutative unital rings. We denote by Spec D the set of all prime ideals of D, endowed with the topology whose closed sets are exactly those of the form Spec(D, X)

= def {P ∈ Spec D | X ⊆ P } , for X ⊆ D.
It can be easily seen that Spec D is a spectral space.

The assignment D → Spec D can be extended to a (contravariant) functor, by sending any 0, 1-lattice homomorphism f :

D → E to the map Spec f : Spec E → Spec D, Q → f -1 [Q].
This map is easily seen to be continuous, and even spectral (cf. Section 2.1). Hence, Spec defines a contravariant functor, from the category of all bounded distributive lattices, with 0, 1-lattice homomorphisms, to spectral spaces, with spectral maps.

Stone duality [START_REF] Marshall | Topological representations of distributive lattices and Brouwerian logics[END_REF] is contained in the following theorem and the subsequent comments.

Theorem 2.4 (Stone). The pair (Spec,

• K) induces a (categorical ) duality, between bounded distributive lattices with 0, 1-lattice homomorphisms and spectral spaces with spectral maps.

The dual of a bounded distributive lattice D is its spectrum Spec D, and the dual of a spectral space X is the lattice • K(X) of all its compact open subsets. The dual of a spectral map ϕ : X → Y , between spectral spaces, is the 0, 1-lattice homomorphism

• K(ϕ) : • K(Y ) → • K(X), V → ϕ -1 [V ]. For a bounded distributive lattice D, the assignment a → {P ∈ Spec D | a / ∈ P } defines a lattice isomorphism ε D : D → • K(Spec D)
, and the assignment D → ε D defines a natural transformation from the identity functor, on bounded distributive lattices, to the functor

• K • Spec.
For a spectral space X, the assignment While Stone's Theorem and Hochster's Theorem both characterize spectral spaces as spectra of commutative rings and bounded distributive lattices, respectively, the former result achieves one more feature: since we are dealing with a duality, every spectral space is the spectrum of a unique (up to isomorphism) bounded distributive lattice, while a commutative unital ring is usually not determined by its Zariski spectrum (e.g., the Zariski spectrum of any field is the one-point topological space).

x → {U ∈ • K(X) | x / ∈ U } defines a homeomorphism η X : X → Spec • K(X),
Something can still be said about the interaction between Stone duality and Hochster's Theorem. For a subset X in a commutative unital ring A, the Zariskiclosed set Spec(A, X) introduced in (2.1) depends only on the ideal I generated by X: namely, Spec(A, X) = Spec(A, I). This can be even made more precise, by observing that Spec(A, I) = Spec(A, √ I), where √ I, the radical of I, is defined as {x ∈ A | x n ∈ I for some positive integer n}. Now this is as precise as it can get, as by Krull's Theorem, every radical ideal (an ideal I is radical if I = √ I) is the intersection of all the prime ideals containing it. Building on that observation, it is easy to verify that the Stone dual of Spec A is isomorphic to the (bounded, distributive) lattice Id r c A of all radicals of finitely generated ideals4 (beware the ambiguity about the "finitely generated radical ideal" terminology!) of A. Hence, by applying Stone duality to Theorem 2.3, we obtain the following.

Corollary 2.5. Every bounded distributive lattice with zero is isomorphic to Id r c A for some commutative unital ring A.

Congruence lattices

For more detail and references about this section, we refer the reader to Wehrung [START_REF] Schmidt | Pudlák's Approaches to CLP, Lattice Theory: Special Topics and Applications[END_REF][START_REF]Congruences of Lattices and Ideals of Rings[END_REF][START_REF]Liftable and Unliftable Diagrams[END_REF].

The Congruence Lattice Problem.

A congruence of a lattice (L, ∨, ∧) is an equivalence relation θ on L such that x 1 ≡ θ y 1 and x 2 ≡ θ y 2 implies both x 1 ∨ x 2 ≡ θ y 1 ∨ y 2 and x 1 ∧ x 2 ≡ θ y 1 ∧ y 2 , whenever x 1 , x 2 , y 1 , y 2 ∈ L (we say that θ is compatible with the operations ∨ and ∧). Here, x ≡ θ y is short for (x, y) ∈ θ.

The set Con L of all congruences of a lattice L, partially ordered under ⊆, is a complete lattice, in which

i∈I θ i = i∈I θ i , i∈I θ i = congruence generated by i∈I θ i , for every collection {θ i | i ∈ I} of congruences of L.
A congruence θ is finitely generated if it is the least one such that x 1 ≡ θ y 1 and

• • • and x n ≡ θ y n , for some x i , y i ∈ L (1 ≤ i ≤ n). A congruence θ is finitely generated iff it is a compact element of Con L, that is, whenever θ ⊆ i∈I θ i , there exists a finite subset J of I such that θ ⊆ i∈J θ i .
The lattice Con L is algebraic, that is, it is complete and every congruence can be written in the form i∈I θ i with all θ i compact.

Up to this point, there is nothing special about the structure of lattice and the concept of congruence can be extended to any "universal algebra" (i.e., nonempty set A with a [possibly infinite] collection of operations A n → A for various n). For example, the congruences of a group G are in one-to-one correspondence with the normal subgroups of G. However, the congruences of a lattice L are, usually, not in any natural one-to-one correspondence with subsets of L.

The congruence lattice of any universal algebra is an algebraic lattice. The converse, stating that every algebraic lattice arises as the congruence lattice of some universal algebra, is a deep theorem of Grätzer and Schmidt [START_REF] Grätzer | Characterizations of congruence lattices of abstract algebras[END_REF].

A large part of what makes lattices so special is the following fundamental result, established in Funayama and Nakayama [START_REF] Funayama | On the distributivity of a lattice of latticecongruences[END_REF]. We show a proof for convenience. Theorem 3.1 (Funayama and Nakayama). The congruence lattice of every lattice is distributive.

Proof. Define the lower median operation on a lattice L by setting m(x, y, z) = def (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z), whenever x, y, z ∈ L. The map m is a majority operation, that is, m(x, x, y) = m(x, y, x) = m(y, x, x) = x whenever x, y ∈ L. Now let α, β, γ ∈ Con L and let (x, y) ∈ α ∩ (β ∨ γ). Since (x, y) belongs to β ∨ γ, which is the transitive closure of β ∪ γ, there exists a finite sequence (z 0 , . . . , z n ) of elements of L such that z 0 = x, z n = y, and each (z i , z i+1 ) ∈ β ∪ γ. Since m is a composition of the fundamental operations ∨ and ∧, every congruence of L is compatible with m. Setting t i = def m(x, y, z i ) for each i, it follows that each (t i , t i+1 ) ∈ β ∪ γ. Moreover, since m is a majority operation, t 0 = x and t n = y. From x ≡ α y it follows that each t i ≡ α m(x, x, z i ) = x, thus each (t i , t i+1 ) ∈ α∩(β ∪γ) = (α∩β)∪(α∩γ). Therefore, (x, y) ∈ (α∩β)∨(α∩γ), thus completing the proof that α∩(β ∨γ) ⊆ (α∩β)∨(α∩γ). The converse containment is trivial. Theorem 3.1 is quite lattice-specific, in the sense that it does not extend to groups, modules, rings. . . For example, usually A ∩ (B + C) = (A ∩ B) + (A ∩ C) for submodules A, B, C of a given module.

In the 1940's, Dilworth proved that conversely, every finite distributive lattice is the congruence lattice of a (finite) lattice. Then he asked whether this could be extended to the infinite case:

The Congruence Lattice Problem (CLP), ∼ 1940. Is every distributive algebraic lattice the congruence lattice of a lattice? CLP initiated a considerable amount of work, leading to a host of positive results. All those results are more conveniently stated in terms of the set Con c L of all compact (i.e., finitely generated) congruences of L, partially ordered under set inclusion. It should be noted that Con c L is not a lattice as a rule: for compact congruences α and β, the join α ∨ β is compact, but the meet α ∩ β may not be compact. Hence, Con c L is a (∨, 0)-semilattice 5 . It is distributive, that is, whenever

α ⊆ β 1 ∨ β 2 in Con c L, there are α i ⊆ β i in Con c L such that α = α 1 ∨ α 2 .
Moreover, one can go naturally from Con L to Con c L (the latter is the semilattice of all compact elements of the former) and back (the former is isomorphic to the ideal lattice of the latter), and this functorially. We thus reach the following Semilattice formulation of CLP. Is every distributive (∨, 0)-semilattice representable, that is, isomorphic to Con c L for some lattice L? 5 Or, equivalently, a commutative, idempotent monoid, endowed with the partial ordering ≤ given by x ≤ y ⇔ x ∨ y = y.

{0, 1} {0, 1} 2 p : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ {0, 1} 2 p O O {0, 1} 2 p d d ■ ■ ■ ■ ■ ■ ■ ■ ■ {0, 1} 2 ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ {0, 1} 2 s d d ■ ■ ■ ■ ■ ■ ■ ■ ■ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ {0, 1} 2 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ {0, 1} e d d ■ ■ ■ ■ ■ ■ ■ ■ ■ e O O e : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ Figure 3.1.
The heavy cube D c (e.g., HK = KH for normal subgroups in a group). However, not every lattice is congruence-permutable (e.g., consider the three-element chain). The following result was established in Růžička, Tůma, and Wehrung [START_REF] Růžička | Distributive congruence lattices of congruence-permutable algebras[END_REF].

Theorem 3.7. The cube D c cannot be lifted (with respect to the functor Con c ), by any cube of congruence-permutable (universal ) algebras. In particular, it cannot be lifted by groups, rings, or modules.

It is still unknown whether every diagram of finite Boolean (∨, 0)-semilattices and (∨, 0)-homomorphisms, indexed by a finite lattice, can be lifted with respect to the functor Con c on lattices.

The proof of Theorem 3.7 (i.e., non-liftability of D c by congruence-permutable algebras), yielded the construction of a distributive (∨, 0)-semilattice that cannot be lifted by congruence-permutable algebras, in Růžička, Tůma, and Wehrung [START_REF] Růžička | Distributive congruence lattices of congruence-permutable algebras[END_REF].

Theorem 3.8. For every cardinal number κ ≥ ℵ 2 , the distributive (∨, 0, 1)-semilattice S κ is not isomorphic to Con c A for any congruence-permutable algebra A. In particular, S ℵ2 is not isomorphic to Con c A whenever A is a sectionally complemented lattice, a group, a module, or a ring. Moreover, in the case of sectionally complemented lattices, groups, modules, rings, the cardinality bound ℵ 2 is optimal.

Since congruences of a module are identified with submodules, it follows, for example, that S ℵ2 is not isomorphic to the submodule lattice of any module. Similarly, it is not isomorphic to the normal subgroup lattice of any group.

In all the results described in this section, the unliftability statements about cubes (of order-dimension n = 3) always parallel unliftability statements about objects of cardinality ℵ n-1 = ℵ 2 . General principles explaining this correspondence are developed in the monograph Gillibert and Wehrung [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF]. Nonetheless, the constructions described in the present section predate the monograph.

Nonstable K 0 -theory

For further details and references about this section, see Ara [START_REF]The realization problem for von Neumann regular rings, Ring theory[END_REF], Goodearl [START_REF] Kenneth | von Neumann regular rings and direct sum decomposition problems, Abelian groups and modules[END_REF], Wehrung [START_REF]Lifting defects for nonstable K 0 -theory of exchange rings and C * -algebras[END_REF].

Two idempotent matrices a and b over a (not necessarily commutative or unital) ring R are Murray -von Neumann equivalent, in symbol a ∼ b, if there are matrices x and y such that a = xy and b = yx. For square matrices x and y over R,

we set x ⊕ y = def x 0 0 y . If x 1 ∼ y 1 and x 2 ∼ y 2 , then x 1 ⊕ x 2 ∼ y 1 ⊕ y 2 ,
V(R) = def {[a] | a idempotent matrix on R} is commutative (x + y = y + x) and conical (x + y = 0 ⇒ x = y = 0). It encodes the nonstable K 0 -theory of R. If R is unital, then V(R)
is isomorphic to the monoid of isomorphism classes of all finitely generated projective right (or, equivalently, left) R-modules, and the enveloping group (also called Grothendieck group) of V(R) is the group usually denoted by K 0 (R).

For example, if R is a field, or more generally a division ring, then the finitely generated right R-modules are exactly the finite-dimensional right R-vector spaces, which are classified by dimension. Hence, V(R) is isomorphic to the monoid N 0 = def {0, 1, 2, . . . } of nonnegative integers.

The following result was established in Bergman [START_REF] Bergman | Coproducts and some universal ring constructions[END_REF], Bergman and Dicks [START_REF] Bergman | Universal derivations and universal ring constructions[END_REF]:

Theorem 4.1 (Bergman and Dicks). Every commutative conical monoid M is isomorphic to V(R M ) for some ring R M .

To my knowledge, the functoriality of the assignment M → R M has not yet been studied.

Further restrictions on the ring R are usually met with restrictions on the nonstable K 0 -theory. An important class of rings for which this occurs is the following. It was introduced by Warfield [29] in the unital case, Ara [START_REF] Ara | Extensions of exchange rings[END_REF] in the general case. Definition 4.2. A ring R is an exchange ring if for all x ∈ R, there are an idempotent e ∈ R and r, s ∈ R such that e = rx = x + s -sx.

It can be established (and this is not trivial) that this condition is left-right symmetric. Every von Neumann regular ring (i.e., satisfying (∀x)(∃y)(xyx = x)) is an exchange ring, and a C*-algebra is an exchange ring iff it has real rank zero (Ara et al. [START_REF] Ara | Separative cancellation for projective modules over exchange rings[END_REF], Ara [START_REF] Ara | Extensions of exchange rings[END_REF]).

Extending earlier results about von Neumann regular rings and C*-algebras of real rank zero, Ara established in [START_REF] Ara | Extensions of exchange rings[END_REF] the following result, in full generality. Theorem 4.3 (Ara). Let R be an exchange ring. Then V(R) is a refinement monoid, that is, for all a 0 , a

1 , b 0 , b 1 ∈ V(R) such that a 0 + a 1 = b 0 + b 1 , there are c i,j ∈ V(R), for i, j ∈ {0, 1}, such that each a i = c i,0 + c i,1 and b i = c 0,i + c 1,i .
It is not known whether every conical refinement monoid M arises in this way, that is, M ∼ = V(R) for some exchange ring R. Nevertheless, it is known since Wehrung [START_REF] Wehrung | Non-measurability properties of interpolation vector spaces[END_REF] that the answer to that problem is negative for von Neumann regular rings (with counterexamples of cardinality ℵ 2 ), and it is known since Wehrung [START_REF]Lifting defects for nonstable K 0 -theory of exchange rings and C * -algebras[END_REF] that the answer is negative for C*-algebras of real rank zero (with counterexamples of cardinality ℵ 3 , using the CLL tool from Gillibert and Wehrung [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF]).

Say that a monoid is simplicial if it is isomorphic to a finite power of the additive monoid N 0 of all nonnegative integers. The following result was established in Wehrung [START_REF]Lifting defects for nonstable K 0 -theory of exchange rings and C * -algebras[END_REF]: Theorem 4.4. There is a commutative cube, of simplicial monoids and monoid homomorphisms, that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by von Neumann regular rings nor by C*-algebras of real rank 0.

The cube of Theorem 4.4 is obtained from the one of Figure 3.1 by replacing {0, 1} by N 0 and setting e(x) = (x, x), p(x, y) = x + y, and s(x, y) = (y, x) whenever x, y ∈ N 0 .

Theorem 4.4 could be paraphrased by saying that at diagram level, the nonstable K 0 -theory of exchange rings properly contains both the one of von Neumann regular rings and the one of C*-algebras of real rank zero. By using again the tools of Gillibert and Wehrung [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF], this result is extended to objects in [START_REF]Lifting defects for nonstable K 0 -theory of exchange rings and C * -algebras[END_REF].

Theorem 4.5. There exists a unital exchange ring of cardinality ℵ 3 (resp., an ℵ 3 -separable unital C*-algebra of real rank 1) R such that V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a von Neumann regular ring.

Back to spectral spaces

For more details and references about the present section, we refer the reader to Delzell and Madden [START_REF]Lattice-ordered rings and semialgebraic geometry. I, Real analytic and algebraic geometry[END_REF], Grätzer [15, § 2.5], Johnstone [18, § II.3 and Ch. V], Keimel [START_REF]Some trends in lattice-ordered groups and rings, Lattice theory and its applications[END_REF], Coste and Roy [START_REF] Coste | La topologie du spectre réel, Ordered fields and real algebraic geometry[END_REF], Dickmann [START_REF] Dickmann | Applications of model theory to real algebraic geometry. A survey[END_REF]Ch. 6], Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF][START_REF]Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF].

The Zariski spectrum construction can be extended to various contexts, such as Abelian ℓ-groups (yielding the ℓ-spectrum) and partially ordered, commutative unital rings (yielding the real spectrum). Tailoring the methods above (in particular, CLL) to that new context, further results can be obtained on ℓ-spectra and real spectra. Let us give a short summary of such results. 5.1. The ℓ-spectrum of an Abelian ℓ-group. An ℓ-group is a group G endowed with a translation-invariant lattice ordering. An ℓ-ideal of G is an order-convex normal subgroup of G, closed under the lattice operations. An ℓ-ideal P of an Abelian ℓ-group G is prime if it is proper (i.e., P = G) and x ∧ y ∈ P implies that either x ∈ P or y ∈ P , whenever x, y ∈ G. The ℓ-spectrum of G, denoted Spec ℓ G, is the set of all prime ℓ-ideals of G, endowed with the topology whose closed sets are exactly those of the form Spec

ℓ (G, X) = def {P ∈ Spec ℓ G | X ⊆ P } , (5.1)
for X ⊆ G. An order-unit of G is an element e of the positive cone G + of G such that every element of G + lies below an integer multiple of e. To ease the presentation, let us deal only with Abelian ℓ-groups with order-unit. For those ℓ-groups, Spec ℓ G is a spectral space. (If G has no order-unit, then Spec ℓ G is only a generalized spectral space.) A spectral space X is completely normal if for all z ∈ X and all x, y in the closure of {z}, either x belongs to the closure of {y} or y belongs to the closure of {x}. It is known since Keimel [19] that the ℓ-spectrum of any Abelian ℓ-group with unit is completely normal. The characterization problem of all ℓ-spectra of Abelian ℓ-groups with unit has been open since then, sometimes under equivalent names (mostly Mundici's MV-spectrum problem, see [START_REF] Mundici | Advanced Lukasiewicz Calculus and MV-Algebras[END_REF]). The countable case was recently solved in Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]: Theorem 5.1. Every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of an Abelian ℓ-group with order-unit.

Due to a counterexample by Delzell and Madden [START_REF] Delzell | A completely normal spectral space that is not a real spectrum[END_REF], Theorem 5.1 cannot be extended to spectral spaces that fail to be second countable. In fact, due to the results of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF], the class of all Stone duals of ℓ-spectra of Abelian ℓ-groups with unit cannot be characterized by any class of L ∞,ω -formulas (thus, in particular, it is not first-order).

5.2. The real spectrum of a commutative unital ring. Let A be a commutative unital ring. A subset C of A is a cone if it is both an additive and a multiplicative submonoid of A, containing all squares in A. A cone P of A is prime if A = P ∪(-P ) and the support P ∩(-P ) is a prime ideal of A. For a prime cone P , -1 / ∈ P (otherwise 1 ∈ P ∩ (-P ), thus P ∩ (-P ) = A, a contradiction). We denote by Spec r A the set of all prime cones of A, endowed with the topology generated by all subsets of the form {P ∈ Spec r A | a / ∈ P }, for a ∈ A, and we call Spec r A the real spectrum of A. The real spectrum of A is always a completely normal spectral space. Delzell and Madden [START_REF] Delzell | A completely normal spectral space that is not a real spectrum[END_REF] established that not every completely normal spectral space arises in this way. Mellor and Tressl [22] extended that result by proving that for any infinite cardinal number λ, the class of Stone duals of all real spectra cannot be defined by any class of L ∞,λ -formulas.

It is not known whether the result of Theorem 5.1 can be extended to real spectra: is every second countable completely normal spectral space the real spectrum of some commutative unital ring?

It is still possible to compare ℓ-spectra, real spectra, and their spectral subspaces, with respect to inclusion. Let for any class X of spectral spaces. (By definition, a spectral space X is a spectral subspace of a spectral space Y if X ⊆ Y and the inclusion map from X into Y is a spectral map.) The following result was established in Wehrung [START_REF]Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF].

Theorem 5.2. All containments and non-containments between the classes CN, ℓ, Sℓ, R, SR can be read on the following diagram:

CN = SCN Sℓ SR ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ℓ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ R ◗ ◗ ◗ ◗ ◗ ◗
Moreover, the topological counterexamples proving the various non-containments in that diagram all have bases of cardinality ℵ 1 , except for the one proving Sℓ CN, which has a basis of cardinality ℵ 2 .

The Stone duals of the topological counterexamples proving the relations ℓ Sℓ, R SR, R ⊆ ℓ are all constructed as condensates of one-arrow diagrams. The counterexample illustrating ℓ ⊆ SR is obtained via a direct construction. The counterexample illustrating Sℓ CN is obtained via a "free construction" similar, in spirit, to the one introduced in Wehrung [START_REF] Wehrung | Non-measurability properties of interpolation vector spaces[END_REF].

Acknowledgment

This paper is a text of the author's plenary talk at the 83th annual conference of the Indian Mathematical Society, which took place in Tirupati (Andhra Pradesh, India) during December 12-15, 2017. My deepest thanks are going to the organizers of the conference for their invitation and outstanding hospitality, which resulted in a week full of sunshine in both physical and metaphorical sense, with special thanks to S. Sreenadh and N. K. Thakare.

  and the assignment X → η X is a natural transformation from the identity functor, on spectral spaces, to the functor Spec • • K. The duality described above restricts to the classical Stone duality between Boolean algebras and zero-dimensional compact Hausdorff spaces. It can also be extended, mutatis mutandis, to what should be called Stone duality between distributive lattices with zero, with cofinal 3 zero-preserving lattice homomorphisms, and so-called "generalized spectral spaces" with spectral maps (cf. Rump and Yang [26, page 63], Johnstone [18, § II.3], Grätzer [15, § II.5]).

  for all square matrices (not necessarily of the same dimension) x 1 , x 2 , y 1 , y 2 . Hence, Murray -von Neumann equivalence classes [a] = def {x | a ∼ x}, for idempotent matrices a over R, can be added, via the rule [a] + [b] = def [a ⊕ b]. The monoid

  of Abelian ℓ-groups with unit} , R = def {real spectra of commutative unital rings} , SX = def {spectral subspaces of members of X} ,

For posets P and Q, a map f : P → Q is cofinal if for every q ∈ Q there exists p ∈ P such that q ≤ f (p).

The letter "r" stands for "radical" while the letter "c" stands for "compact", which is the lattice-theoretical counterpart of "finitely generated".

Some known positive instances of CLP are given by the following result: Theorem 3.2. Let S be a distributive (∨, 0)-semilattice. In each of the following cases, S is representable:

(i) S is countable (Bauer ∼ 1980);

(ii) card S ≤ ℵ 1 (Huhn 1989); (iii) S is a lattice (Schmidt 1981);

(iv) S = lim -→n<ω S n , with all transition maps S n → S n+1 (∨, 0)-homomorphisms and all S n distributive lattices (Wehrung 2003).

In any of those cases, a representing lattice L (such that Con c L ∼ = S) can be taken sectionally complemented (a lattice L with zero is sectionally complemented if whenever a ≤ b in L, there exists x ∈ L such that a ∨ x = b and a ∧ x = 0). The following result was established in Wehrung [START_REF] Wehrung | Non-measurability properties of interpolation vector spaces[END_REF][START_REF]A uniform refinement property for congruence lattices[END_REF].

Theorem 3.3. For every cardinal number κ ≥ ℵ 2 , there exists a distributive (∨, 0, 1)-semilattice S κ , of cardinality κ, not isomorphic to Con c L for any sectionally complemented lattice L.

The task of removing "sectionally complemented" from the statement of Theorem 3.3 was completed nearly ten years later in Wehrung [START_REF]A solution to Dilworth's congruence lattice problem[END_REF], thus yielding a negative solution of CLP, using the class of counterexamples constructed for Theorem 3.3 (with a far more elaborate proof):

Theorem 3.4. The distributive (∨, 0, 1)-semilattice S ℵω+1 is not representable.

The optimal cardinality bound was subsequently obtained by Růžička in [START_REF] Růžička | Free trees and the optimal bound in Wehrung's theorem[END_REF]: Theorem 3.5 (Růžička). The distributive (∨, 0, 1)-semilattice S ℵ2 is not representable.

Those results, together with the proof of Theorem 3.1, shift the original Congruence Lattice Problem to the following (still unsolved) problem: is every distributive algebraic lattice the congruence lattice of a majority algebra? By definition, a majority algebra is a pair (A, m), where m : A 3 → A is a majority operation. The proof of Theorem 3.1 shows that the congruence lattice of any majority algebra is distributive.

3.2.

A heavy cube, and congruence-permutable algebras. Referring to the terminology "lifting" (with respect to a functor) from Section 1, "heavy" is intended to mean "hard to lift".

We consider the diagram D c of (∨, 0)-semilattices and (∨, 0)-homomorphisms represented in Figure 3.1, where e(x) = (x, x), p(x, y) = x ∨ y, and s(x, y) = (y, x) whenever x, y ∈ {0, 1}. This diagram is obviously commutative.

The following result was established in Tůma and Wehrung [START_REF] Tůma | Simultaneous representations of semilattices by lattices with permutable congruences[END_REF].

Theorem 3.6. The cube D c cannot be lifted (with respect to the functor Con c ), by any cube of sectionally complemented lattices and lattice homomorphisms.

In fact, it turns out that Theorem 3.6 can be extended to a much broader algebraic context; in particular, it is not lattice-specific. For binary relations α and β on a set A, we set α • β = def {(x, y) ∈ A × A | (∃z ∈ A)((x, z) ∈ α and (z, y) ∈ β)}. We say that an algebra A is congruence-permutable if α• β = β • α for all congruences α and β of A. For example, groups, modules, rings are all congruence-permutable