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It is well known that the lattice Idc G of all principal {\ell}-ideals of any Abelian {\ell}-group G is a completely normal distributive 0-lattice, and that not every completely normal distributive 0-lattice is a homomorphic image of some Idc G, via a counterexample of cardinality $\aleph 2. We prove that every completely normal distributive 0-lattice with at most $\aleph 1 elements is a homomorphic image of some Idc G. By Stone duality, this means that every completely normal generalized spectral space, with at most $\aleph 1 compact open sets, is homeomorphic to a spectral subspace of the {\ell}-spectrum of some Abelian {\ell}-group.

International audience; We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and counterexamples, also highlighting the particular role played by \emph{positive} orderability.Moreover, we construct a non-left-orderable, positively right-orderable submonoid of the free product of the cyclic group of order 7 with the free group on four generators.Any group extension of that monoid has elements of order 7.

We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x $\in$ D | a $\le$ b $\lor$ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x $\le$ y $\lor$(x-y),(x-y)$\land$(y-x) = 0, and x-z $\le$ (x-y)$\lor$(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex {\ell}-subgroups of any (not necessarily Abelian) {\ell}-group. It has $\aleph 2 elements. This solves negatively a few problems stated by Iberkleid, Mart{\'i}nez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal $\lambda$, the class of Stone duals of spectra of all Abelian {\ell}-groups with order-unit is not closed under L $\infty$$\lambda$-elementary equivalence., Comment: 23 pages. v2 removes a redundancy from the definition of a Cevian operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on the G side)

v4 is the final version; International audience; In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Extending our original machinery, about finite lattices of polyhedra, from linear to affine and allowing relativizations to convex subsets, then invoking Baro's Normal Triangulation Theorem, we obtain the following result:Theorem. For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian {\ell}-group (we say that D is {\ell}-representable) iff for all a, b $\in$ D there are x, y $\in$ D such that a $\lor$ b = a $\lor$ y = b $\lor$ x and x $\land$ y = 0. On the other hand, we construct a non-{\ell}-representable bounded distributive lattice, of cardinality $\aleph$ 1 , with an {\ell}-representable countable L$\infty, \omega$-elementary sublattice. In particular, there is no characterization, of the class of all {\ell}-representable distributive lattices, in arbitrary cardinality, by any class of L$\infty, \omega$ sentences., Comment: Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee).In Corollary 8.6, X, Y^-, and Y^+ are just elements of \Op(\mathcal{H}) (not necessarily basic open)

In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups., 27 pages

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form [Formula: see text]. We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex [Formula: see text]-subgroups of members of any class of [Formula: see text]-groups containing all Archimedean [Formula: see text]-groups; the class of all semilattices of finitely generated [Formula: see text]-ideals of members of any nontrivial quasivariety of [Formula: see text]-groups; the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem; the class of all semilattices of finitely generated two-sided ideals of rings; the class of all semilattices of finitely generated submodules of modules; the class of all monoids encoding the nonstable K0-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero; (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large [Formula: see text]-frame. The main underlying principle is that under quite general conditions, for a functor [Formula: see text], if there exists a noncommutative diagram [Formula: see text] of [Formula: see text], indexed by a common sort of poset called an almost join-semilattice, such that [Formula: see text] is a commutative diagram for every set [Formula: see text], [Formula: see text] for any commutative diagram [Formula: see text] in [Formula: see text], then the range of [Formula: see text] is anti-elementary.

The interval monoid $\Upsilon$(P) of a poset P is defined by generators [x, y], where x $\le$ y in P , and relations [x, x] = 1, [x, z] = [x, y] $\times$ [y, z] for x $\le$ y $\le$ z. It embeds into its universal group $\Upsilon$ $\pm$ (P), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results: $\bullet$ The monoid $\Upsilon$(P) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice). $\bullet$ For every group G, there is a poset P of length 2 such that $\Upsilon$(P) is a gcd-monoid and G is a free factor of $\Upsilon$ $\pm$ (P) by a free group. Moreover, P can be taken finite iff G is finitely presented. $\bullet$ For every finite poset P , the monoid $\Upsilon$(P) can be embedded into a free monoid. $\bullet$ Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid Umon(S) of any category S. This enables us, in particular, to characterize the embeddability of Umon(S) into a group, by stating that it holds at the hom-set level. We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group. We illustrate our results by various examples and counterexamples., Comment: 27 pages (v4). Semigroup Forum, to appear

International audience; It is well-known that the weak Bruhat order on the symmetric group on a finite number n of letters is a lattice, denoted by P(n) and often called the permutohedron on n letters, of which the Tamari lattice A(n) is a lattice retract. The equational theory of a class of lattices is the set of all lattice identities satisfied by all members of that class. We know from earlier work that the equational theory of all P(n) is properly contained in the one of all A(n). We prove the following results. Theorem I. The equational theory of all P(n) and the one of all A(n) are both decidable. Theorem II. There exists a lattice identity that holds in all P(n), but that fails in a certain 3338-element lattice. Theorem III. The equational theory of all extended permutohedra, on arbitrary (possibly infinite) posets, is trivial. In order to prove Theorems I and II, we reduce the satisfaction of a given lattice identity in a Cambrian lattice of type A to a certain tiling problem on a finite chain. Theorem I then follows from Büchi's decidability theorem for the monadic second-order theory MSO of the successor function on the natural numbers. It can be extended to any class of Cambrian lattices of type A with MSO-definable set of orientations.

24 pages.Misprints v1: In the Abstract, the last (4) should be (5). In Proposition 4.3, spectral --> generalized spectral.Main change from v1 to v2: the observation that real spectra = unital Brumfiel spectra.; It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We prove the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. (5) There exists a completely normal spectral space which cannot be embedded , as a spectral subspace, into any ℓ-spectrum. The commutative unital rings and Abelian lattice-ordered groups in (2), (3), (4) all have cardinality ℵ 1 , while the spectral space of (5) has a basis of cardinality ℵ 2. Moreover, (3) solves a problem by Mellor and Tressl.

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence)., Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added, typos corrected; final version due to appear in Journal of Combinatorial Algebra

While Theorem 4.8.9 implies that the type monoid of a Boolean inverse semigroup S can be any countable conical refinement monoid, there are situations in which the structure of S impacts greatly the one of \(\mathop{\mathrm{Typ}}\nolimits S\). A basic illustration of this is given by the class of AF inverse semigroups , introduced in Lawson and Scott [77], which is the Boolean inverse semigroup version of the class of AF C*-algebras. Another Boolean inverse semigroup version of a class of C*-algebras, which we will not consider here, is given by the Cuntz inverse monoids studied in Lawson and Scott [77, § 3].

Tarski investigates in [109] partial commutative monoids constructed from partial bijections on a given set. In Kudryavtseva et al. [71], this study is conveniently formalized in the language of inverse semigroups. Further connections can be found in works on K-theory of rings, such as Ara and Exel [7].

The type monoid of a Boolean inverse semigroup is an abstraction of the concept of monoid of equidecomposability types of a Boolean ring under a group action. The latter concept has been studied in a wide array of works including Banach [17], Tarski [109]. Its relation with type monoids of Boolean inverse semigroups was recognized in Wallis’ Ph.D. thesis [116], see also Kudryavtseva et al. [71], Lawson and Scott [77].

The axioms of ring theory, when deprived of the existence of additive inverses, yield the axioms of semirings. When endowed with an additional involutary anti-automorphism (we will talk about involutary semirings), semirings will enjoy quite a fruitful interaction with Boolean inverse semigroups, the basic idea being to have the multiplications agree and the inversion map correspond to the involution.

We can find in the literature many proposals for generalizations of permutohedra. Among those, let us mention the permutohedron on a poset (Pouzet et al. [356]), multinomial lattices (also called lattices of multipermutations, see Bennett and Birkhoff [55], Flath [154], Santocanale [393]), lattices of generalized permutations (Gross [210], Krob et al. [288], Boulier et al. [82]).

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids 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 regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring., Comment: 34 pages. Algebras and Representation Theory, to appear

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\kappa$ elements., Comment: Fund. Math., to appear

A partial lattice P is ideal-projective, with respect to a class C of lattices, if for every K $\in$ C and every homomorphism $\phi$ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f : P $\rightarrow$ K for $\phi$ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to C. We prove the following: (1) A finite lattice P, belonging to a variety V, is sharply transferable with respect to V iff it is projective with respect to V and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to V. (2) Every finite distributive lattice is sharply transferable with respect to the class R mod of all relatively complemented modular lattices. (3) The gluing D 4 of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety V iff V is contained in the variety M$\omega$ generated by all lattices of length 2. (4) D 4 is projective, but not ideal-projective, with respect to R mod. (5) D 4 is transferable, but not sharply transferable, with respect to the variety M of all modular lattices. This solves a 1978 problem of G. Gr\"atzer. (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson., Comment: Theorem 3.9(e) is redundant (it is contained in Theorem 3.9(a))To appear in Order

International audience; We find a distributive (v, 0, 1)-semilattice S of size $\\aleph_1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: - There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to S. - There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to S. These results are established by constructing an infinitary statement, denoted here by URPsr, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice S.

For a positive integer n, we denote by SUB (respectively, SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (respectively, P of length at most n). We prove the following results:(1) SUBnis a finitely based variety, for any n≥1.(2) SUB2is locally finite.(3) A finite atomistic lattice L without D-cycles belongs to SUB if and only if it belongs to SUB2; this result does not extend to the nonatomistic case.(4) SUBnis not locally finite for n≥3.

We investigate categorical and amalgamation properties of the functor Idc assigning to every partially ordered abelian group G its semilattice of compact ideals Idc G. Our main result is the following. Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the Idc functor, by a diagram of pseudo-simplicial vector spaces. Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces (over the rationals) with interpolation. The methods introduced make it also possible to prove the following ring-theoretical result. Theorem 2. For any countable distributive join-semilattices S and T and any field K, any (v,0)-homomorphism $f: S\to T$ can be lifted, with respect to the Idc functor on rings, by a homomorphism $f: A\to B$ of K-algebras, for countably dimensional locally matricial algebras A and B over K. We also state a lattice-theoretical analogue of Theorem 2 (with respect to the Conc functor, and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively

We say that a (∨,0)-semilattice S is conditionally co-Brouwerian, if (1) for all nonempty subsets X and Y of S such that X $\leq$ Y (i.e., x $\leq$ y for all (x, y) $\in$ X x Y), there exists z $\in$ S such that X $\leq$ z $\leq$ Y, and (2) for every subset Z of S and all a, b $\in$ S, if a $\leq$ b ∨ z for all z $\in$ Z, then there exists c $\in$ S such that a $\leq$ b ∨ c and c $\leq$ Z. By restricting this definition to subsets X, Y, and Z of less than $\kappa$ elements, for an infinite cardinal $\kappa$, we obtain the definition of a conditionally $\kappa$- co-Brouwerian (∨, 0)-semilattice. We prove that for every conditionally co-Brouwerian lattice S and every partial lattice P, every (∨, 0)-homomorphism $\phi$: Conc P $\to$ S can be lifted to a lattice homomorphism f : P $\to$ L, for some relatively complemented lattice L. Here, Conc P denotes the (∨, 0)-semilattice of compact congruences of P. We also prove a two-dimensional version of this result, and we establish partial converses of our results and various of their consequences in terms of congruence lattice representation problems. Among those consequences, for every infinite regular cardinal $\kappa$ and every conditionally $\kappa$-co-Brouwerian S of size $\kappa$, there exists a relatively complemented lattice L with zero such that Conc L is isomorphic to S.