Types of polynomial completeness of expanded groups

The polynomial functions of an algebra preserve all congruence relations. In addition, if the algebra is finite, they preserve the labelling of the congruence lattice in the sense of Tame Congruence Theory. The question is for which algebras every congruence preserving function, or at least every function that preserves the labelling of the congruence lattice, is a polynomial function. In this paper, we investigate this question for finite algebras that have a group reduct. Presented by K. Kaarli. Received March 12, 2006; accepted in final form October 16, 2008. The second author is supported by Grant No. 144011 of the Ministry of Science of the Republic of Serbia, and the Scholarship “One-Month Visits to Austria for University Graduates” WUS-Austria, from the Austrian Ministry of Education, Science and Culture.     

The inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

On the (un)decidability of a near-unanimity term

We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem. While working on this paper, the author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant nos. T 37877 and K 60148.     

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.     

Infinite highly connected planar graphs of large girth

We construct infinite planar graphs of arbitrarily large connectivity and girth, and study their separation properties. These graphs have no thick end but continuum many thin ones. Every finite cycle separates them, but they corroborate Diestel’s conjecture that everyk-connected locally finite graph contains a possibly infinite cycle — see [3] — whose deletion leaves it (k — 3)-connected.     

Cancellation in entropic algebras

We describe the equational theory of the class of cancellative entropic algebras of a fixed type. We prove that a cancellative entropic algebra embeds into an entropic polyquasigroup, a natural generalization of a quasigroup. In fact our results are even more general and some corollaries hold also for non-entropic algebras. For instance an algebra with a binary cancellative term operation, which is a homomorphism, is quasi-affine. This gives a strengthening of K. Kearnes’ theorem. Our results generalize theorems obtained earlier by M. Sholander and by J. Ježek and T. Kepka in the case of groupoids. The work on this paper was conducted within the framework of INTAS project no. 03 51 4110 “Universal algebra and lattice theory”. The author was also supported by the Statutory Grant of Warsaw University of Technology no. 504G11200013000.     

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

Equational theories as congruences of enriched monoids

The problem of characterizing the lattices of equational theories is still unsolved. In this paper we describe a class of monoids enriched by two unary operations and show that a lattice L is a lattice of equational theories if and only if L is isomorphic to a lattice of congruences of some enriched monoid belonging to . The author was supported by INTAS grant 03-51-4110 and The Alexander von Humboldt Foundation.     

On left conjugacy closed loops with a nucleus of index two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop. Work supported by institutional grant MSM 113200007 and by Grant Agency of Czech Republic, Grant 201/02/0594. The paper was written while the author was visiting Universitaet Hamburg in January 2004.     

On tensor products of completely positive linear maps between pro-C *-algebras

In this paper we define the tensor products of completely positive linear maps between pro-C^*-algebras and discuss about connection between the KSGNS construction associated with the strict completely positive linear maps ρ and θ and the KSGNS construction associated with ρ ⊗ θ. This research was partially supported by CEEX grant -code PR-D11-PT00-48/2005 from The Romanian Ministry of Education and Research and partially by CNCSIS (Romanian National Council for Research in High Education) grant-code A 1065/2006.     

On left conjugacy closed loops in which the left multiplication group is normal

A loopQ is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. This papers investigates those LCC loops where the group generated by left translations is normal in the group generated by both left and right translations.     

The product of two von Neumann n-frames, its characteristic, and modular fractal lattices

Let L be a bounded lattice. If for each a₁ < b₁ ∈ L and a₂ < b₂ ∈ L there is a lattice embedding ψ: [a₁, b₁] → [a₂, b₂] with ψ(a₁) = a₂ and ψ(b₁) = b₂, then we say that L is a quasifractal. If ψ can always be chosen to be an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals, then L will be called a fractal lattice. For a ring R with 1 let denote the lattice variety generated by the submodule lattices of R-modules. Varieties of this kind are completely described in [16]. The prime field of characteristic p will be denoted by F_p. Let be a lattice variety generated by a nondistributive modular quasifractal. The main theorem says that is neither too small nor too large in the following sense: there is a unique , a prime number or zero, such that and for any n ≥ 3 and any nontrivial (normalized von Neumann) n-frame of any lattice in , is of characteristic p. We do not know if in general; however we point out that, for any ring R with 1, implies . It will not be hard to show that is Arguesian. The main theorem does have a content, for it has been shown in [2] that each of the is generated by a single fractal lattice L_p; moreover we can stipulate either that L_p is a continuous geometry or that L_p is countable. The proof of the main theorem is based on the following result of the present paper: if is a nontrivial m-frame and is an n-frame of a modular lattice L with m, n ≥ 3 such that and , then these two frames have the same characteristic and, in addition, they determine a nontrivial mn-frame of the same characteristic in a canonical way, which we call the product frame. Presented by E. T. Schmidt.     

Terms which are Mal’cev modulo some functions

We derive consequences from the existence of a term which satisfies Mal’cev identities (characterizing permutability) modulo two functions F and G from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal’cev term modulo F and G. Received September 21, 2006; accepted in final form March 20, 2007.     

Avoidable structures, II: Finite distributive lattices and nicely structured ordered sets

Let be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in that are well-quasi-ordered by embeddability, and thus characterize the members of that belong to at least one infinite anti-chain in . While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.     

Avoidable structures, I: Finite ordered sets, semilattices and lattices

We find all finite unavoidable ordered sets, finite unavoidable semilattices and finite unavoidable lattices. While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates. Presented by R. Willard.