Terms that are self-dual in Boolean algebras but not in MO 2

An absorption law is an identity of the form p = x. The ternary function x+y+z (ring addition) in Boolean algebras satisfies three absorption laws in two variables. If a term satisfies these three identities in a variety, it is called a minority term for that variety. We construct a minority term p for orthomodular lattices such the identity defines Boolean algebras modulo orthomodular lattices. (The dual of p is denoted by .) Consequently, having a unique minority term function characterizes Boolean algebras among orthomodular lattices. Our main result generalizes this example to arbitrary arity and arbitrary consistent sets of 2-variable absorption laws. Presented by J. Berman.     

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples. Received June 28, 2005; accepted in final form January 23, 2007.     

2-uniform congruences in majority algebras and a closure operator

A ternary term m(x, y, z) of an algebra is called a majority term if the algebra satisfies the identities m(x, x, y) = x, m(x, y, x) = x and m(y, x, x) = x. A congruence α of a finite algebra is called uniform if all of its blocks (i.e., classes) have the same number of elements. In particular, if all the α-blocks are two-element then α is said to be a 2-uniform congruence. If all congruences of A are uniform then A is said to be a uniform algebra. Answering a problem raised by Gr?tzer, Quackenbush and Schmidt [2], Kaarli [3] has recently proved that uniform finite lattices are congruence permutable. In connection with Kaarli’s result, our main theorem states that for every finite algebra A with a majority term any two 2-uniform congruences of A permute. Examples show that we can say neither “algebra” instead of “algebra with a majority term”, nor “3-uniform” instead of “2-uniform”. Given two nonempty sets A and B, each relation gives rise to a pair of closure operators, which are called the Galois closures on A and B induced by ρ. Galois closures play an important role in many parts of algebra, and they play the main role in formal concept analysis founded by Wille [4]. In order to prove our main theorem, we introduce a pair of smaller closure operators induced by ρ. These closure operators will hopefully find further applications in the future. Dedicated to the memory of Kazimierz Głazek Presented by E. T. Schmidt. Received November 29, 2005; accepted in final form May 23, 2006. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T049433 and T037877.     

Endomorphisms of monadic Boolean algebras

A classical result about Boolean algebras independently proved by Magill [10], Maxson [11], and Schein [17] says that non-trivial Boolean algebras are isomorphic whenever their endomorphism monoids are isomorphic. The main point of this note is to show that the finite part of this classical result is true within monadic Boolean algebras. By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadic Boolean algebras the endomorphism monoid of each of which has only one element (namely, the identity), this being the first known example of a variety that is not universal (in the sense of Hedrlín and Pultr), but contains a proper class of non-isomorphic rigid algebras (that is, the identity is the only endomorphism). Received February 3, 2006; accepted in final form September 5, 2006.     

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.     

Poset algebras over well quasi-ordered posets

A new class of partial order-types, class is defined and investigated here. A poset P is in the class iff the poset algebra F(P) is generated by a better quasi-order G that is included in L(P). The free Boolean algebra F(P) and its free distributive lattice L(P) have been defined in [ABKR]. The free Boolean algebra F(P) contains the partial order P and is generated by it: F(P) has the following universal property. If B is any Boolean algebra and f is any order-preserving map from P into a Boolean algebra B, then f can be extended to a homomorphism of F(P) into B. We also define L(P) as the sublattice of F(P) generated by P. We prove that if P is any well quasi-ordering, then L(P) is well founded, and is a countable union of well quasi-orderings. We prove that the class is contained in the class of well quasi-ordered sets. We prove that is preserved under homomorphic image, finite products, and lexicographic sum over better quasi-ordered index sets. We prove also that every countable well quasi-ordered set is in . We do not know, however if the class of well quasi-ordered sets is contained in . Additional results concern homomorphic images of posets algebras. The third author was supported by the following institutions: Israel Science Foundation (postdoctoral positions at Ben Gurion University 2000–2002), The Fields Institute (Toronto 2002–2004), and by The Nato Science Fellowship (University Paris VII, CNRS-UMR 7056, 2004).     

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.     

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”.     

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.     

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.     

Congruences on equivalence algebras

A certain class of atomic, semimodular, semisimple partition lattices is studied. It is shown that this class is precisely the class of congruence lattices of equivalence algebras. The first author is granted by project POCTI-ISFL-1-143 of the “Centro de álgebra da Universidade de Lisboa”, supported by FCT and FEDER.     

A generalized flat extension theorem for moment matrices

In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.      

Subdirect products of hereditary congruence lattices

A congruence lattice L of an algebra A is called power-hereditary if every 0-1 sublattice of Lⁿ is the congruence lattice of an algebra on Aⁿ for all positive integers n. Let A and B be finite algebras. We prove

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.     

Skew representations of twisted Yangians

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL _N and the Yangian for $$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians $$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras $$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of the twisted Yangian $$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms $$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over $$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.     

Invariants of simple algebras

We determine the group of invariants with values in Galois cohomology with coefficients of central simple algebras of degree at most 8 and exponent dividing 2. The work of A. S. Merkurjev has been supported by the NSF grant DMS #0652316.     

Applications of a theorem of Singerman about Fuchsian groups

Assume that we have a (compact) Riemann surface S, of genus greater than 2, with , where is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to where is a Fuchsian group such that and has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions. Received: 4 April 2008     

Advertibly complete locallym-convex two-sided algebras

Many results on two-sided Banach algebras remain valid form-convex normal ones; in particular, the commutativity modulo the Jacobson radical. Moreover advertible completeness appears to be sufficient.     

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.     

Undecidability of the centers of groups and group algebras

Let k ≧ 3 be an integer or k = ∞ and let K be a field. There is a recursive family of finitely presented groups G_n over a fixed finite alphabet with solvable word problem such that