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We study partial orders defined on the set of points of space-time that are invariant under Lorentz transformations. Kirszbraun's Theorem allows to show that world lines of particles evolving in space-time are precisely the maximal chains in the causality order. We show that the causality order is well behaved in the sense that it is gradable and level sets under various gradings are precisely the anti-chain cutsets. We also show that the causality orders corresponding to different light speed parameters c are essentially the only partial orders invariant under Lorentz transformations and under some other, more obvious affine transformations of space-time. We characterize optical lines and hyperplanes, inertial lines and planes, and separation lines as well, in terms of the causality order and use these characterizations to provide a variant proof of the Alexandrov-Zeeman Theorem. [ABSTRACT FROM AUTHOR]

Intervals in binary or n-ary relations or other discrete structures generalize the concept of an interval in a linearly ordered set. They are defined abstractly as closed sets of a closure system on a set, satisfying certain axioms. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions. This result is used to show that the lattice of interval decompositions is balanced, and the case when this lattice is distributive is also characterised.

We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our characterization replaces the condition of gradedness with a second modularity condition on the maximal chain of left-modular elements. [ABSTRACT FROM AUTHOR]

Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality and restrictions of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. It is shown that for semilattices the semimodularity ensures the good behaviour of the distances considered. The Jordan-Dedekind chain condition, which is weaker than semimodularity, is equivalent to the basic criterion that the graphtheoretic distance (realized by zig-zagging up and down freely in the poset to connect two points) is compatible with distances measured on chains by the relative height. Semimodularity is shown to be equivalent to the validity of the triangle inequality of a restricted graph-theoretic distance, called the up-down distance. The fact that the up-down distance corresponds to the computation of degrees of kinship in family trees leads to the observation that the less familiar canon-law method of computation corresponds also to a mathematically well behaved Chebyshev-type distance on discrete semilattices. For the Chebyshev distance also semimodularity is shown to imply the validity of the triangle inequality. The reverse implication fails, but assuming the validity of the triangle inequality, the semimodularity is shown to have a local characterization by a forbidden six-element subsemilattice. Like in the classical case of real spaces, the Chebyshev semilattice distance is shown to be the limit of a converging sequence of distances, all of them verifying the triangle inequality if the semilattice is semimodular. [ABSTRACT FROM AUTHOR]

The Huffman coding algorithm is interpreted in the context of the lattice of partitions of the source alphabet. Maximal chains in the partition lattice correspond to linear extensions of tree orders, and those among the chains that exhibit a simple greedy property correspond precisely to executions of the Huffman algorithm. [ABSTRACT FROM AUTHOR]

An alternative proof is given of the existence of greatest lower bounds in the imbalance order of binary maximal instantaneous codes of a given size. These codes are viewed as maximal antichains of a given size in the infinite binary tree of 0–1 words. The proof proposed makes use of a single balancing operation within the same imbalance poset of codes of the same fixed size, instead of moving back and forth between posets of codes corresponding to two different code sizes using expansion and contraction, as in the previous proofs of the existence of glb. It also makes use of a new combinatorial characterization of the imbalance order. [ABSTRACT FROM AUTHOR]

Intervals in binary or n-ary relations or other discrete structures generalize the concept of an interval in a linearly ordered set. They are defined abstractly as closed sets of a closure system on a set, satisfying certain axioms. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions. This result is used to show that the lattice of interval decompositions is balanced, and the case when this lattice is distributive is also characterised. [ABSTRACT FROM AUTHOR]

A selection of ring theory papers by Jenő Szigeti is reviewed with an emphasis on aspects related to matrix algebras. [ABSTRACT FROM AUTHOR]

The Cayley graph construction provides a natural grid structure on a finite vector space over a field of prime or prime square cardinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distance from the null vector in the grid graph defines a Manhattan norm. The Hermitian inner product on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz inequality is valid with respect to the Manhattan norm. With respect to the non-transitive order provided by the quadratic residue tournament, an analogue of the Cauchy-Schwarz inequality holds in arbitrarily large neighborhoods of the null vector, when the characteristic is an appropriate large prime. [ABSTRACT FROM AUTHOR]

Associating to each pre-order on the indices 1,.....,n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n×n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relations. Certain conjugates of the upper or lower triangular matrix rings correspond to the various linear orderings of the indices, and the incidence algebras of partial orderings arise as intersections of such conjugate matrix rings. [ABSTRACT FROM AUTHOR]

An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, Subrings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068-1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subrings Mn(θ, R) of the full n × n matrix ring Mn(R) over a ring R, which are closed with respect to taking inverses. It turns out that Mn(θ, R) is closed with respect to taking inverses in Mn(R) if all the equivalence classes with respect to θ ∩ θ-1, except possibly one, are of a size less than or equal to p (say) and Mp(R) is Dedekind finite. Another purpose of this article is to show that Mn(θ, R) is Dedekind finite if and only if Mm(R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ-1). This provides a positive result for the inheritance of Dedekind finiteness by a matrix ring (albeit not a full matrix ring) from a smaller (full) matrix ring. [ABSTRACT FROM AUTHOR]

Abstract: We examine classes of real-valued functions of 0–1 variables closed under algebraic operations as well as topological convergence, and having a certain local characteristic (requiring that any function not in the class should have a -variable minor not belonging to this class). It is shown that for , the only 4 maximal classes with these properties are those of submodular, supermodular, monotone increasing and monotone decreasing functions. All the 13 locally defined closed classes are determined and shown to be intersections of the 4 maximal ones. All maximal classes for are determined and characterized by the sign of higher order derivatives of the functions in the class. [Copyright &y& Elsevier]

Finite Lorentz groups acting on four-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real four-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary precision to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields. [ABSTRACT FROM AUTHOR]

The classes of Boolean functions closed under classical compositions form an algebraic lattice which was completely described in 1941 in a pioneering work of Post (Ann. Math. Stud. (5) (1941)). These classes and the lattice are often referred to as the Post Classes and the Post Lattice, respectively. There are several approaches to present a Post class. Being a set of functions it can be characterized by a traditional set-theoretic description of its members. Since the Post classes are closed under certain operations, they are also often presented by sets of generators. A remarkable approach, which has been widely used in Universal Algebra is a characterization of classes by means of preservation of polyrelations (Kibernetika (3)(Pt1) (1969) 1, (5)(PtII) (1969) 1). Recently, there appeared several new methods for the characterization of classes of logic functions. These methods are based on special formal expression, which in general define a much larger variety of classes particularly including all Post classes. One such result is by Ekin et al. (Discrete Math. 211 (2000) 27), which presents the characterization of classes by a set (possibly infinite) of certain equational identities. The approach developed in [Ekin et al. (2000)] was soon extended to several directions. Pippenger (Discrete Math. 254 (2002) 405) presented the classes through pairs of relations (constrains) in the setting of a Galois Theory. Pogosyan (Multiple Valued Logic, Gordon and Breach, London, 2001, pp. 417–448, Vol. 7) has defined each such class by one functional term (possibly of infinite length), and has introduced the notion of rank for a term as well as for a class. In (Algebra Universalis 44 (2000) 309) established a connection with the Birkhoff-Tarski HSP Theorem. This paper presents a complete characterization of the Post Classes by means of functional terms (as in [Pogosyan, 2001]). We also give a constructive criterion which establishes the minimal ranks for all Post classes. [Copyright &y& Elsevier]