Weakly representable relation algebras form a variety

We prove that the class of weakly representable relation algebras is closed under homomorphic images, hence it is a variety. As a corollary we classify the subdirectly irreducible algebras in this class. Received April 3, 2007; accepted in final form February 7, 2008.     

Many-valued quantum algebras

We deal with algebras of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety. Supported by the Research and Development Council of the Czech Government via the project MSM6198959214.     

On the variety question for wRRA

In this paper we consider a question of Jónsson [6] whether the class of weakly representable relation algebras is a variety. We prove that the class is closed under taking homomorphic images provided that a certain embedding condition obtains. Received June 21, 2005; accepted in final form October 17, 2006.     

Amalgamation,interpolation and epimorphisms in algebraic logic

In his landmark paper on amalgamation published in Algebra Universalis in 1971, Don Pigozzi posed some open questions in connection with amalgamation of subclasses of cylindric algebras. Some of these questions were originally raised by Comer, Daigneault, Johnson, McKenzie and others. In this paper we give answers to all these as well as a number of other related questions. Most of the solutions were found by the authors of this paper. However, a few were contributed by others who will of course be given due credit at the appropriate points. This paper is dedicated to Don Pigozzi on his retirement. Presented by R. W. Quackenbush. The first author’s research was supported by the Hungarian National Foundation for Scientific Research grants no T30314 and T23234. The second author’s research was supported by the Hungarian National Foundation for Scientific Reserach OKTA grant no T30314. All the new results in this article were announced by Judit Madarász in the Workshop on Abstract Algebraic Logic, held in Centre de Recerca Matematica Bellaterra, Spain July 1–5, 1997. Received December 15, 2002; accepted in final form January 3, 2006.     

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.     

Orthocomplemented lattices with a symmetric difference

Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices (). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that “live” on set-representable OMLs. Received April 10, 2007; accepted in final form February 12, 2008.     

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”.     

Perfect IFG-Formulas

IFG logic is a variant of the independence-friendly logic of Hintikka and Sandu. We answer the question: “Which IFG-formulas are equivalent to ordinary first-order formulas?” We use the answer to prove the ordinary cylindric set algebra over a structure can be embedded into a reduct of the IFG-cylindric set algebra over the structure.      

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.     

Bimodules associated to vertex operator algebras

Let V be a vertex operator algebra and m, n ≥ 0. We construct an A _n (V)-A _m (V)-bimodule A _n,m (V) which determines the action of V from the level m subspace to level n subspace of an admissible V-module. We show how to use A _n,m (V) to construct naturally admissible V-modules from A _m (V)-modules. We also determine the structure of A _n,m (V) when V is rational. Chongying Dong was supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz. Cuipo Jiang was supported in part by China NSF grant 10571119.     

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.     

The Surabale of Space-Time and Its Algebra Signature Split

A “surable” is a category given by a special manifold of geometric algebra frames. It is a bale brought on by a surjective map the equivalence classes of which can constitute base elements of the associative algebra. It is also a stranded braid of idempotents based on a sheaf of base unipotents. The stranded braid of idempotents which are thought to represent particles or fields consists of fibers strictly separated by mutual annihilation throughout the bundle. The surabale can be defined on the Clifford algebra of space-time. Then it constitutes a bundle of frames which – though covering all dimensions of the geometric algebra – turn out as isomorphic to the ground space generating the algebra. Because of this, the mass shell and the Dirac-Hestenes equation can be defined on the whole surabale. As a result the equation is preserved when acted on by the symmetries of the transformation group of the standard model. The K?hler-equation simply turns into a Dirac-Hestenes equation on the inhomogenous surabale, yet with the same simple differential 1-forms of the linear equation. This shows very beautifully that the equation of motion as well as the invariance of the surabale under the standard model symmetry can be formulated base free. The Clifford bases – instead of Gra?mann – just brings in the riches of representation, that is, the emergence of the standard model. But its grading, in a way, is an illusion. Studying the dimension of the space-time-split in quadratic Clifford algebras, it turns out that the dimension of the positive space-like component reproduces their period-8 properties. Considering as an example the Minkowski space-time in the Lorentz metric rather than in (+  −   −   − ) we can see that physicists found the electroweak symmetries in the negative part of the geometry, here denoted as , but did not realize the strong force symmetries in the positive part since those depend on the graded motion in the graded subspace . It is comparatively difficult to find the generators of the group capable to represent the classic SU(3) with its root space A₂. Though the approach put forward gives satisfying answers to some classical problems of relativistic quantum mechanics, it does not solve the most important riddle which has been variously pointed out by Professor Oziewicz, namely, mechanics is not governed by the Lorentz- or Poincaré-group. The simplest argument to be held against it, is that the Lorentz/Poincaré group by definition is the symmetry group of the metric tensor in the empty space-time without bodies and radiation. How can such a Bewegungsgruppe of the empty space-time be related to the physics and mechanics of material bodies? [1], [2] May be this first argument is not convincing enough. But Oziewicz has listed a considerable number of arguments concerning the whole observation process against the present day unquestioned but incorrect application of the full Lorentz group. To clarify this will still need some more fundamental efforts which do not concern the main subject of this paper.     

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.     

Group gradings on diagonal algebras

We describe all group gradings on the diagonal algebra k ⁿ , where k is an arbitrary field. Received: 21 January 2008     

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.     

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.     

Peak point theorems for uniform algebras on smooth manifolds

It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if each point of X is a peak point for A, then A = C(X). This peak point conjecture was disproved by Brian Cole in 1968. However, Anderson and Izzo showed that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth two-manifolds with boundary. The corresponding assertion for smooth three-manifolds is false, but Anderson, Izzo, and Wermer established a peak point theorem for polynomial approximation on real-analytic three-manifolds with boundary. Here we establish a more general peak point theorem for real-analytic three-manifolds with boundary analogous to the two-dimensional result. We also show that if A is a counterexample to the peak point conjecture generated by smooth functions on a manifold of arbitrary dimension, then the essential set for A has empty interior.