Finite Members Research Articles

The structure of finite commutative idempotent involutive residuated lattices

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.

Orthomodular lattices that are horizontal sums of Boolean algebras

The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $\mathcal H$ of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by $\mathcal H$ but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class $\mathcal H$ in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of $\mathcal H$.

Regular double p-algebras

Abstract In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDP n , n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP 1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP 1, LV (RDP 1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP 1 and conclude that LV (RDP 1) is countably infinite. An equational basis for each proper subvariety of RDP 1 is given. To study the subvarieties RDP n with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2 ℵ 0 . We also use Priestley duality to prove that RDP and each of its subvarieties RDP n are generated by their finite members.

The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokic have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the variety of distributive lattices is not an exception, but an instance of a more general phenomenon. We show that for almost all nontrivial locally finite varieties of lattices no “reasonable” expansion of the finite members of the variety by linear orders gives rise to a Ramsey class. The responsibility for this lies not with the lattices as structures, but with the lack of algebraic morphisms: if we consider lattices as partially ordered sets (and thus switch from algebraic embeddings to embeddings of relational structures) we show that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property. It now comes as no surprise that the same is true for varieties of semilattices.

Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A_Gamma)).

The subpower membership problem for bands

Fix a finite semigroup S and let a1,…,ak,b be tuples in a direct power Sn. The subpower membership problem (SMP) for S asks whether b can be generated by a1,…,ak. For bands (idempotent semigroups), we provide a dichotomy result: if a band S belongs to a certain quasivariety, then ▪ is in P; otherwise it is NP-complete.Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable ▪. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.

The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L A via the quasi-order → on the finite members of the variety generated by A, where B →C if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L A induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as L Q , for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕1, where L is an interval in the subgroup lattice of a finite group.

Commutative idempotent groupoids and the constraint satisfaction problem

A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol–Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, \({x(x(yz)) \approx (x(xy))z}\)). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol–Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law \({x(yz) \approx (xy)(xz)}\), and show that they are tractable.

Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the \({\{\rightarrow, \forall,1\}}\)-subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by \({\mathcal{ML}}\), and we give an equational basis for this variety. An algebra in \({\mathcal{ML}}\) is called a monadic Łukasiewicz implication algebra. We characterize the subdirectly irreducible members of \({\mathcal{ML}}\) and the congruences of every monadic Łukasiewicz implication algebra by monadic filters. We prove that \({\mathcal{ML}}\) is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

Research on Application of Non-Equilateral Angle Steel for Transmission Towers

In Chinese transmission towers, all the steels are equilateral angle, most of the diagonal and secondary members are in one-sided bolt connection working condition, so the force is eccentric. When the length within a certain range, relative to the equilateral angle, non-equilateral angle steel under this working condition to be able to withstand a larger load, and the ability to cross-sectional area smaller, reducing the weight of the whole base tower. In order to solve the above problems, through research and analysis of large-scale finite element software ANSYS diagonal and secondary members for transmission tower angle steel model length which can be equilateral angle replaced non-equilateral angle steel. Eventually found that when the length is small long side is connected to non-equilateral angle steel are available, and when the length is bigger short side is connected to non-equilateral angle steel are available.

The finite model property for semilinear substructural logics

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.

Relation algebras as expanded FL-algebras

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.

Minimal Varieties of Representable Commutative Residuated Lattices

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17---19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms and two integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members.

The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

Abstract We prove the result announced by the title as well as some of its consequences.

The decidability of some classes of Stone algebras

Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses \({{\bf SA}^{\rm i}_n}\) and \({{\bf SA}^{\rm s}_n}\) of the class SA n of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class of all Post algebras of degree n is definitionally equivalent to the intersection \({{\bf SA}^{\rm i}_{n} \cap {\bf SA}^{\rm s}_{n}}\). We show that for each n ≥ 2 the class \({{\bf SA}^{\rm i}_n}\) is hereditarily undecidable while \({{\bf SA}^{\rm s}_{n}}\) is decidable. As a consequence we obtain several (un)decidability results for various axiomatic classes of Stone algebras: among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA n is decidable.

Pre-ideals of Basic Algebras

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order. In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.

Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.

Asymptotic classes of finite Moufang polygons

In this paper we study the model theory of classes of finite Moufang polygons. We show that each family of finite Moufang polygons forms an ‘asymptotic class’. As a result, since every non-principal ultraproduct of an asymptotic class is ‘measurable’, and therefore supersimple of finite rank, we obtain examples of (infinite) supersimple Moufang polygons of finite rank. In a forthcoming paper, [8], we will show that all supersimple Moufang polygons of finite rank arise over supersimple fields and belong to exactly those families which also have finite members. This body of work will give a description of groups with supersimple finite rank theory which have a definable spherical Moufang BN-pair of rank at least two.

Residual properties of pre-bipartite digraphs

We say that a directed graph is pre-bipartite if its symmetric closure is bipartite. We will show that the class \({\mathcal{B}}\) of all pre-bipartite digraphs containing no cycles is a universal Horn class. Let \({\mathcal{U}}\) be a universal Horn class contained in \({\mathcal{B}}\). We determine when it is possible to axiomatise, by first-order sentences, the class \({\mathcal{R}_{\rm CT}(\mathcal{U}_{\rm fin})}\) of compact topological digraphs that are topologically residually in the class of finite members of \({\mathcal{U}}\). We show that if \({\mathcal{R}_{\rm CT}(\mathcal{U}_{\rm fin})}\) is axiomatisable by first-order sentences, then it is axiomatisable by universal Horn sentences.