Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈G, a 〉 \G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates ag = g−1ag of a by elements g ∈ G generate a semigroup denoted 〈ag | g ∈ G〉. We classify the finite permutation groups G on a finite set...

Mitsch’s natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also descri...

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Hölder continuous drift term. We prove existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Itô-Tanaka type. The proof requires non-standard elliptic estimates in Hölder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type fo...

This paper studies FA-presentable structures and gives a complete classification of the finitely generated FA-presentable cancellative semigroups. We show that a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group.

Let P be a finite biprefix code and let >t: = (9, S) be the syntactic transformation semigroup fts:r of P’. We show that if e ES is an idenpotent, then the ts X, = (Qe, eSe) CORS~~+S of partial one to one maps. We also show that any ts of partial one to one maps divides a ts ci partial one fto one maps which is the syntactic ts of a finite biprefix code.

Abstract. We show that the Camassa–Holm equation ut −uxxt +3uux −2uxuxx −uuxxx = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u|t=0 in H. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measur...

Fix a finite semigroup S and let a1, . . . , ak , b be tuples in a direct power S. The subpower membership problem (SMP) for S asks whether b can be generated by a1, . . . , ak. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups wit...

For a set X , an equivalence relation ρ on X , and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and imageR. We determine the str...

where the unknown function/is from a real number interval into a Banach space X. For suitable real numbers t and vectors x in X, Ait, x) is the infinitesimal generator of a holomorphic semigroup of linear contraction operators in X, and certain regularity requirements are placed on the function (/, x) -> Ait, x). After proving a local existence, uniqueness, and stability theorem for (*), we con...

We discuss the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exact...