The monoids of the patience sorting algorithm

The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed. [Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

A Categorization Theory of Spatial Voting: How the Center Divides the Political Space

This article presents a categorization theory of spatial voting, which postulates that voters perceive political stances through coarse classifications. Because voters think in terms of categories defined by the ideological center, their behavior deviates from standard models of utility maximization along ideological continua. Their preferences are characterized by discontinuities, rewarding parties on their side of the ideological space more than existing spatial models would predict. While this study concurs with prior studies suggesting that voters tend to use a proximity rule, it argues that this rule mainly serves to distinguish among parties of the same side. Overall, the results suggest that voters’ party evaluations are characterized by a nontrivial identity component, generating in-group biases not captured by the existing spatial models of voting.

IDENTITIES AND A BOUNDED HEIGHT CONDITION FOR SEMIGROUPS

We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

Identities in the universal envelopes of Lie algelras

It is well known (see Latyšev [1] for finite dimensional case) that the universal envelope of Lie Alaebra g over a commutative field ť of characteristic 0 is a PI-alaebra (i.e. possesses a nontrivial identity) if and only if this Lie algebra is abelian. On the other hand the recent results due to Passman [2] describe the conditions under which the group algebra of a group over an arbitrary commutative field is a PI-algebra. A. L. Šmel'kin suggested that I should find necessary and sufficient conditions for a Lie algebra g over a field of nonzero characteristic under which its universal envelope Ug should be a PI-algebra. These conditions are given in the following theorem.

Cubic identity graphs and planar graphs derived from trees

The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.