On relative ranks of finite transformation semigroups with restricted range

UDC 512.5 We determine the relative rank of the semigroup of all transformations on a finite chain with restricted range modulo the set of all orientation-preserving transformations in Moreover, we state the relative rank of the semigroup modulo the set of all order-preserving transformations in In both cases we characterize the minimal relative generating sets.

Characterizations of Cayley graphs of finite transformation semigroups with restricted range

The transformation semigroup with restricted range [Formula: see text] is the set of all functions from a set [Formula: see text] into a non-empty subset [Formula: see text] of [Formula: see text]. In this paper, we characterize Cayley graphs of [Formula: see text] with the connection set [Formula: see text]. Moreover, the undirected property of Cayley graphs Cay [Formula: see text] is studied.

Symmetry Analysis, Bäcklund Transformations, and Interaction Solutions for an AB Modified KdV System

An AB modified KdV (AB-mKdV) system which can be used to describe two-place event is studied in this manuscript. Because the AB-mKdV system is considered as a special reduction of the famous AKNS system, the properties of the AKNS system are first revealed by using symmetry analysis. The nonlocal symmetries related to truncated Painlevé expansion, the finite transformation, and the symmetry reduction solutions of the AKNS system are presented. The corresponding Bäcklund transformations and the interaction solutions of the AB-mKdV system are constructed based on the special reduction. The results demonstrate that the AB-mKdV system possesses many kinds of interaction solutions, such as the interactions between kink and soliton and kink and cnoidal waves. The soliton can be changed from bright to dark during propagation.

The complexity of properties of transformation semigroups

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in [Formula: see text]. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in [Formula: see text]. Deciding whether a semigroup has a left (respectively, right) zero is shown to be [Formula: see text]-complete, as are the problems of testing whether a transformation semigroup is nilpotent, [Formula: see text]-trivial or has central idempotents. We also give [Formula: see text] algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in [Formula: see text]. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is [Formula: see text]-complete.

Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.