We consider those positive integers that are not representable as linear combinations of terms of a generalized arithmetic progression with nonnegative integer coefficients. To do this, we make use of the numerical semigroup generated by a generalized arithmetic progression. The number of integers nonrepresentable by such a numerical semigroup is determined as well as that of its dual. In addit...

We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a most general (group-valued or semigroup-v...

In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the imaginary axis and the norm of its resolvent operator is polynomially bounded near these points. We characterize classes of perturbations preserving the strong stabi...

Let T X be the full transformation semigroup on the setX and let T X,Y {α ∈ T X : Xα ⊆ Y}. Then T X,Y is a sub-semigroup of T X determined by a nonempty subset Y of X. In this paper, we give a necessary and sufficient condition for T X,Y to be regular. In the case that T X,Y is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green’s re...

In a semigroup, the combinatorial definitions of syndetic, piecewise syndetic, and IP are equivalent to their algebraic characterizations in terms of βS. We introduce the analogous definitions and characterizations of syndetic, piecewise syndetic, and IP for an adequate partial semigroup and show that equivalence between the combinatorial definition and algebraic characterization is lost once w...

Denote by (R, ·) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R, ◦) represent R when viewed as a semigroup via the circle operation x ◦ y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R,...

A locally testable semigroup S is a semigroup with the property that for some nonnegative integer k, called the order or level of local testability, two words u and v in some set of generators for S are equal in the semigroup if (1) the pre x and su x of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem fo...

A new fundamental solution semigroup for operator differential Riccati equationsis developed. This fundamental solution semigroup is constructed via an auxiliary finite horizonoptimal control problem whose value functional growth with respect to time horizon is determinedby a particular solution of the operator differential Riccati equation of interest. By exploiting semi-convex...

Fix a finite semigroup S and let a1, . . . , ak , b be tuples in a direct power S. The subpower membership problem (SMP) asks whether b can be generated by a1, . . . , ak. If S is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in nk. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 o...

The construction by Hall of a fundamental orthodox semigroup WB from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup SB that plays the role of WB for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S) ...