Polák theorem and $$\mathbf {}$$-relation on varieties of completely regular semigroups

Completely regular semigroups \(\mathcal {C}\mathcal {R}\) are unions of their subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). The Polák theorem concerns the computation of joins in \(\mathcal {L}(\mathcal {C}\mathcal {R})\). The \(\mathbf {B}\)-relation on \(\mathcal {L}(\mathcal {C}\mathcal {R})\) identifies varieties with the same bands. We elaborate upon two nontrivial conditions in Polák’s theorem applied to certain subsets of \(\mathcal {C}\mathcal {R}\) which amounts to solving particular equations in \(\mathcal {L}(\mathcal {C}\mathcal {R})\).     

Is the Principle of Contradiction a Consequence of $$x^{2}=x$$?

According to Boole it is possible to deduce the principle of contradiction from what he calls the fundamental law of thought and expresses as \(x^{2}=x\). We examine in which framework this makes sense and up to which point it depends on notation. This leads us to make various comments on the history and philosophy of modern logic.     

Identities determining varieties of semigroups with completely regular power

We provide a transparent syntactic algorithm to decide whether an identity defines a variety of semigroups with completely regular power.     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

The Brill–Noether curve and Prym-Tyurin varieties

We prove that the Jacobian of a general curve C of genus $g=2a+1$ , with $a\ge 2$ , can be realized as a Prym-Tyurin variety for the Brill–Noether curve $W^{1}_{a+2}(C)$ . As consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series $g^{a-1}_{3a-2}$ .     

The Ornstein–Uhlenbeck bridge and applications to Markov semigroups

For an arbitrary Hilbert space-valued Ornstein–Uhlenbeck process we construct the Ornstein–Uhlenbeck bridge connecting a given starting point x$x$ and an endpoint y$y$ provided y$y$ belongs to a certain linear subspace of full measure. We derive also a stochastic evolution equation satisfied by the OU bridge and study its basic properties. The OU bridge is then used to investigate the Markov transition semigroup defined by a stochastic evolution equation with additive noise. We provide an explicit formula for the transition density and study its regularity. These results are applied to show some basic properties of the transition semigroup. Given the strong Feller property and the existence of invariant measure we show that all L^p$L^p$ functions are transformed into continuous functions, thus generalising the strong Feller property. We also show that transition operators are q$q$-summing for some q>p>1$q>p>1$, in particular of Hilbert–Schmidt type.     

Classes of semigroups modulo Green’s relation \mathcal{H}

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation \(\mathcal{H}\) , or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.     

The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ 2 matrices

Let \(T_n(\mathbb {F})\) and \(UT_n(\mathbb {F})\) be the semigroups of all upper triangular \(n\times n\) matrices and all upper triangular \(n\times n\) matrices with 0s and/or 1s on the main diagonal over a field \(\mathbb {F}\) with \(\mathsf {char}(\mathbb {F})=0\), respectively. In this paper, we address the finite basis problem for \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety \(\mathsf {var} UT_2(\mathbb {F})\) generated by \(UT_2(\mathbb {F})\) and the involution monoid variety \(\mathsf {var} T_2(\mathbb {F})\) generated by \(T_2(\mathbb {F})\). Moreover, \(\mathsf {var} UT_2(\mathbb {F})\) cannot be defined within \(\mathsf {var} T_2(\mathbb {F})\) by any finite set of identities.     

Perturbations and projections of Kalman–Bucy semigroups

We analyse various perturbations and projections of Kalman–Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman–Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose–Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.     

Semigroup C⁎-algebras and amenability of semigroups

We construct reduced and full semigroup C^?-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C^?-algebras in analogy to the group case.     

Modularity of Calabi-Yau varieties and conformal field theory

We study the modularity problem of Calabi-Yau varieties from the conformal field theo- retic point of view.We express the modular forms associated to all 1-dimensional Calabi-Yau orbifolds in terms of products of Dedekind eta functions,which is hoped to shed light on the modularity questions for higher dimensional Calabi-Yau varieties.     

The structure of finite monoids satisfying the relation ℛ = ℋ

It is shown that any finite monoid S on which Green’s relations R and H coincide divides the monoid of all upper triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid S, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying R = H with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite R-trivial monoids.     

Quasi-ideal transversals of abundant semigroups and spined products

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.     

The Gearhart-Prüss theorem for a class of degenerate semigroups of operators

We consider a class of semigroups of operators in Hilbert space whose generators are linear relations. An analog of the Gearhart-Prüss theorem for semigroups of operators from the class under consideration is obtained.