The VT-operator Semigroup for Two Kinds of Regular Semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. For every rho element of Con S there exists a greatest congruence rhoV [smallest congruence rhov] on S such that the idempotent (rhoV/rho)-classes [(rho/rhov)-classes] are rectangular bands, and a greatest congruence rhoT [smallest congruence rhot] on S such that the idempotent (rhoT/rho-classes [(rho/rhot-classes] are groups. The subsemigroup of the transformation semigroup on Con S generated by the transformation rho → rhoV, rho → rhov, rho → rhoT, and rho → rhot, rho element of Con S, is investigated for orthodox semigroups and cryptogroups. It is shown that in this case this so-called Vt-operator semigroup Omega(S) contains 17 elements at most. A 17-element Vt-operator semigroup Omega(F) is realized for some regular orthogroup F.     

On Embeddability of Idempotent Separating Extensions of Inverse Semigroups

Let S be an inverse semigroup and rho an idempotent separating congruence on S. It is proved that S can be embedded into a lambda-semidirect product of a group F by S/rho where F belongs to the variety generated by the idempotent classes of rho.     

Centralizers and Central Idempotents of Semigroup Rings

. Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r] R\mid ra=ar \quadfor all\quad a] A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s] S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b] S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.     

Standard commutator formula

Let R be a commutative ring with 1, A, B ? R be its ideals, GL(n, R, A) be the principal congruence subgroup of level A in GL(n, A), and E(n, R, A) be the relative elementary subgroup of level A. We prove the following commutator formula

$[E(n,R,A),GL(n,R,B)] = [E(n,R,A),E(n,R,B)],$

which generalizes known results. The proof is yet another variation on the theme of decomposition of unipotents.

     

Minimal Presentations and Efficiency of Semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

The lattice of congruence lattices of algebras on a finite set

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice \(\mathcal {E}\). We describe the atoms and coatoms. Each meet-irreducible element of \(\mathcal {E}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice \(\mathcal {E}\); in particular, we prove that \(\mathcal {E}\) is tolerance-simple whenever \(|A|\ge 4\).     

Meet-irreducible congruence lattices

The system of all congruences of an algebra (A, F) forms a lattice, denoted \({{\mathrm{Con}}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set A forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set A. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubíková-Studenovská et al. (2017). In this paper, we prove necessary and sufficient conditions under which \({{\mathrm{Con}}}(A, f)\) is meet-irreducible in the case when (A, f) is an algebra with short tails (i.e., f(x) is cyclic for each \(x \in A\)) and in the case when (A, f) is an algebra with small cycles (every cycle contains at most two elements).     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

A quadratic approximation for protein sequence to structure mapping

The existence and representations of some generalized inverses, includingA _{T, *} ⁽²⁾ ,A _{T, *} ^(1,2) ,A _{T, *} ^(2,3) ,A _*,S ⁽²⁾ ,A _*,S ^(1,2) andA _*,S ^(2,4) , are showed. As applications, the perturbation theory for the generalized inverseA _T,S ⁽²⁾ and the perturbation bound for unique solution of the general restricted systemAx=b (dim (AT)=dimT,b∈AT andx∈T) are studied. Moreover, a characterization and representation of the generalized inverseA _{T, *} ^Emphasis>(2) is obtained.     

On the semigroup BetaS

Let S be an infinite, discrete, cancellative semigroup and let BetaS be the Stone-Cech compactification of S. Then BetaS is a semigroup with an operation which extends that of S and which is continous only in one variable. We generalize some algebraic properties known to hold for the additive semigroup of the integers.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.     

Monoids for Which Condition (P) Acts are Projective

A characterisation of monoids for which all right S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.     

On the concept of level for subgroups of SL2 over arithmetic rings

We define the concept of level for arbitrary subgroups Γ of finite index in the special linear group SL₂(A _S), whereA _S is the ring ofS-integers of a global fieldk provided thatk is an algebraic number field, or card (S) ≥ 2. It is shown that this concept agrees with the usual notion of ‘Stufe’ for congruence subgroups. In the case SL₂(O),O the ring of integers of an imaginary quadratic number field, this criterion for deciding whether or not an arbitrary subgroup of finite index is a congruence subgroup is used to determine the minimum of the indices of non-congruence subgroups.     

Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.     

The <Emphasis Type="Italic">H</Emphasis>-Closed Monoreflections,Implicit Operations,and Countable Composition,in Archimedean Lattice-Ordered Groups with Weak Unit

In the category of the title, called W, we completely describe the monoreflections \(\mathcal {R}\) which are H-closed (closed under homomorphic image) by means of epimorphic extensions S of the free object on ω generators, F(ω), within the Baire functions on \(\mathbb {R}^{\omega }\), \(B(\mathbb {R}^{\omega })\); label the inclusion \(e_{S} : F(\omega ) \rightarrow S\). Then (a) inj e _S (the class of objects injective for e _S ) is such an \(\mathcal {R}\), with e _S a reflection map iff S is closed under countable composition with itself (called ccc), (b) each such \(\mathcal {R}\) is inj e _S for a unique S with ccc, and (c) if S has ccc, then A∈inj e _S iff A is closed under countable composition with S. We think of (c) as expressing: A is closed under the implicit operations of W represented by S (and these are of at most countable arity). In particular, the family of H-closed monoreflections is a set, whereas the family of all monoreflections is consistently a proper class. There is a categorical framework to the proofs, valid in any sufficiently complete category with free objects and epicomplete monoreflection β which is H-closed and of bounded arity; in W the β is of countable arity, and \(\beta F(\omega ) = B(\mathbb {R}^{\omega })\). The paper continues our earlier work along similar lines.