On uniformly repetitive semigroups

Letk be an integer greater than 1 andS be a finitely generated semigroup. The following propositions are equivalent: 1) the semigroup of non negative integers is not uniformlyk-repetitive; 2) any finitely generated and uniformlyk-repetitive semigroup is finite. As a consequence we prove that any finitely generated and uniformly 4-repetitive semigroup is finite.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

The Gröbner ring conjecture in one variable

We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Gröbner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.     

Amalgamation bases for the class of lattice-ordered groups

We prove Theorem A. The cardinal product of two copies of the integers is an amalgamation base for the class of all lattice-ordered groups but their lexicographic product is not. This answers Problem 27 of [Black Swamp Problem Book (W. Charles Holland, ed.), Bowling Green State University, 1982]. We also prove Theorem B. he cardinal product of n copies of the integers is not an amalgamation base for the class of all lattice-ordered groups if n ≥ 3.     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

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 ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? 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 ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

On the condensation points of the lagrange spectrum

For a positive integerN, L(N) denotes the set of Lagrange values of all sequences (a _k:k=0, ±1, ±2,…) of positive integers with lim sup _k ^ak=N. It is shown that for anyN≥3L(N) has infinitely many condensation points. Such points can be realized as Markov values of symmetric doubly periodic sequences whose period consists of a semi-symmetric tuple.     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.     

Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety $\mathcal{COM}$ of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within $\mathcal{COM}$ by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Je?ek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ??prototypes?? of the first and the third our results.     

Stably free cancellation for abelian group rings

We show that if Γ is a finitely generated abelian group, then every stably free module over Z[Γ] is free.     

Primitive words,free factors and measure preservation

Let F _k be the free group on k generators. A word w ∈ F _k is called primitive if it belongs to some basis of F _k . We investigate two criteria for primitivity, and consider more generally subgroups of F _k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ F _k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F _k is primitive. Again let w ∈ F _k and consider the word map w: G × … × G → G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G × … × G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation, and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and, in particular, prove the conjecture for k = 2. It was asked whether the primitive elements of F _k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F ₂.     

Triangular irreducibility of congruences in quasivarieties

Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer ${m \geqslant 3}$ and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.     

A note on profinite completions and canonical extensions

J. Harding has proved that the profinite limit of an algebra A in a finitely generated variety of monotone lattice expansions coincides with its canonical extension. In this note we drop the monotonicity of the additional operations and prove the same result.     

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

For a positive integer k, let k?+?k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k?≥?1, there exists a constant c _k so that First Fit will use at most $c_kw^2$ chains in partitioning a poset P of width at most w, provided the poset excludes k?+?k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w ^16logw ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead’s exponential bound: (5 ^w ???1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek–Krawczyk–Szczypka bound for the performance of First Fit to 8(k???1)² w, which in turn yields the modest improvement to O(w ^14logw ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t?=?t(k,w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k?+?k as a subposet.     

MacNeille completion and profinite completion can coincide on finitely generated modal algebras

Following Bezhanishvili and Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if ${\mathbb{A}}$ is a residually finite, finitely generated modal algebra such that HSP( ${\mathbb{A}}$ ) has equationally definable principal congruences, then the profinite completion of ${\mathbb{A}}$ is isomorphic to its MacNeille completion, and ? is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra.