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Let
$T_{n}(\mathbb{F})$
be the semigroup of all upper triangular
$n\times n$
matrices over a field
$\mathbb{F}$
. Let
$UT_{n}(\mathbb{F})$
and
$UT_{n}^{\pm 1}(\mathbb{F})$
be subsemigroups of
$T_{n}(\mathbb{F})$
, respectively, having
$0$
s and/or
$1$
s on the main diagonal and
$0$
s and/or
$\pm 1$
s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that
$UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$
and
$T_{2}(\mathbb{F})$
as involution semigroups under the skew transposition are nonfinitely based for any field
$\mathbb{F}$
.
Consider the action of
$\operatorname{GL}(n,\mathbb{Q}_{p})$
on the
$p$
-adic unit sphere
${\mathcal{S}}_{n}$
arising from the linear action on
$\mathbb{Q}_{p}^{n}\setminus \{0\}$
. We show that for the action of a semigroup
$\mathfrak{S}$
of
$\operatorname{GL}(n,\mathbb{Q}_{p})$
on
${\mathcal{S}}_{n}$
, the following are equivalent: (1)
$\mathfrak{S}$
acts distally on
${\mathcal{S}}_{n}$
; (2) the closure of the image of
$\mathfrak{S}$
in
$\operatorname{PGL}(n,\mathbb{Q}_{p})$
is a compact group. On
${\mathcal{S}}_{n}$
, we consider the ‘affine’ maps
$\overline{T}_{a}$
corresponding to
$T$
in
$\operatorname{GL}(n,\mathbb{Q}_{p})$
and a nonzero
$a$
in
$\mathbb{Q}_{p}^{n}$
satisfying
$\Vert T^{-1}(a)\Vert _{p}<1$
. We show that there exists a compact open subgroup
$V$
, which depends on
$T$
, such that
$\overline{T}_{a}$
is distal for every nonzero
$a\in V$
if and only if
$T$
acts distally on
${\mathcal{S}}_{n}$
. The dynamics of ‘affine’ maps on
$p$
-adic unit spheres is quite different from that on the real unit spheres.
Let R be a Mori domain with complete integral closure
$\widehat R$
, nonzero conductor
$\mathfrak f = (R: \widehat R)$
, and suppose that both v-class groups
${{\cal C}_v}(R)$
and
${{\cal C}_v}(3\widehat R)$
are finite. If
$R \mathfrak f$
is finite, then the elasticity of R is either rational or infinite. If
$R \mathfrak f$
is artinian, then unions of sets of lengths of R are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of v-noetherian monoids.
Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.
This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.
The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.
The direct product
$\mathbb{N}\times \mathbb{N}$
of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for
$\mathbb{N}\times S$
, where
$S$
is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if
$S$
is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of
$S$
has a relative left or right identity element.
First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem.
We say that two elements of a group or semigroup are
$\Bbbk$
-linear conjugates if their images under any linear representation over
$\Bbbk$
are conjugate matrices. In this paper we characterize
$\Bbbk$
-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field
$\Bbbk$
.
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup
$T$
into a two-sided semidirect product whose components are built from two subsemigroups
$T_{1}$
,
$T_{2}$
, which together generate
$T$
, and the subsemigroup generated by their setwise product
$T_{1}T_{2}$
. In this sense we decompose
$T$
by merging the subsemigroups
$T_{1}$
and
$T_{2}$
. More generally, our technique merges semigroup homomorphisms from free semigroups.
Let C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.
If H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.
This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called
$\mathbb{F}_{1}$
, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category
$\text{Sch}_{{\mathcal{T}}}$
comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for
${\mathcal{G}}$
in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of
${\mathcal{G}}$
defines a model
$G$
in
$\text{Sch}_{{\mathcal{T}}}$
whose Weyl extension is the Weyl group
$W$
of
${\mathcal{G}}$
. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.
The equational complexity function
$\beta \nu \,:\,{\open N} \to {\open N}$
of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting
$O(\log_{2}^{3}(n))$
growth among varieties of semilattice-ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.
As generalizations of inverse semigroups, Ehresmann semigroups are introduced by Lawson and investigated by many authors extensively in the literature. In particular, Lawson has proved that the category of Ehresmann semigroups and admissible morphisms is isomorphic to the category of Ehresmann categories and strongly ordered functors, which generalizes the well-known Ehresmann–Schein–Nambooripad (ESN) theorem for inverse semigroups. From a varietal perspective, Ehresmann semigroups are derived from reducts of inverse semigroups. In this paper, inspired by the approach of Jones [‘A common framework for restriction semigroups and regular
$\ast$
-semigroups’, J. Pure Appl. Algebra216 (2012), 618–632], Ehresmann semigroups are extended from a varietal perspective to pseudo-Ehresmann semigroups derived instead from reducts of regular semigroups with a multiplicative inverse transversal. Furthermore, motivated by the method used by Gould and Wang [‘Beyond orthodox semigroups’, J. Algebra368 (2012), 209–230], we introduce the notion of inductive pseudocategories over admissible quadruples by which pseudo-Ehresmann semigroups are described. More precisely, we show that the category of pseudo-Ehresmann semigroups and (2,1,1,1)-morphisms is isomorphic to the category of inductive pseudocategories over admissible quadruples and pseudofunctors. Our work not only generalizes the result of Lawson for Ehresmann semigroups but also produces a new approach to characterize regular semigroups with a multiplicative inverse transversal.
Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety
$\mathbf{B}$
of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety
$\mathbf{B}_{R}$
of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices
${\mathcal{L}}(\mathbf{B})$
and
${\mathcal{L}}(\mathbf{B}_{R})$
of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval
$[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$
, every subvariety of
$\mathbf{B}$
is induced from a member of
$\mathbf{B}_{R}$
and vice versa. Here
$\mathbf{D}$
is generated by the three-element left restriction semigroup
$D$
and
$\mathbf{M}$
is the variety of monoids. The analogues hold for pseudovarieties.
Fix a finite semigroup
$S$
and let
$a_{1},\ldots ,a_{k},b$
be tuples in a direct power
$S^{n}$
. The subpower membership problem (SMP) for
$S$
asks whether
$b$
can be generated by
$a_{1},\ldots ,a_{k}$
. 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 with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all
$n$
by
$n$
matrices over a field for
$n\geq 2$
.
We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.
We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids
$L_{\ell }^{1}$
, obtained by adjoining an identity element to the semigroup generated by two idempotents
$a$
and
$b$
with the relation
$0=abab\cdots \,$
(length
$\ell$
). We show that every monoid
$M$
which generates a variety containing
$L_{5}^{1}$
and is contained in the variety generated by
$L_{\ell }^{1}$
for some
$\ell \geq 5$
is nonfinitely based. We establish this result by analysing
$\unicode[STIX]{x1D70F}$
-terms for
$M$
, where
$\unicode[STIX]{x1D70F}$
is a certain nontrivial congruence on the free semigroup. We also show that if
$\unicode[STIX]{x1D70F}$
is the trivial congruence on the free semigroup and
$\ell \leq 5$
, then the
$\unicode[STIX]{x1D70F}$
-terms (isoterms) for
$L_{\ell }^{1}$
carry no information about the nonfinite basis property of
$L_{\ell }^{1}$
.
The author has previously associated to each commutative ring with unit
$R$
and étale groupoid
$\mathscr{G}$
with locally compact, Hausdorff and totally disconnected unit space an
$R$
-algebra
$R\,\mathscr{G}$
. In this paper we characterize when
$R\,\mathscr{G}$
is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.
The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the
$\text{Gl}_{n}(\mathbb{K})$
action on the
$n\times n$
matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in
$M_{n}(\mathbb{K})$
. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.
Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. They may be defined by a simple set of identities and the author initiated a study of the lattice of varieties of such semigroups, in parallel with the study of the lattice of varieties of two-sided restriction semigroups. In this work we study the subvariety
$\mathbf{B}$
generated by Brandt semigroups and the subvarieties generated by the five-element Brandt inverse semigroup
$B_{2}$
, its four-element restriction subsemigroup
$B_{0}$
and its three-element left restriction subsemigroup
$D$
. These have already been studied in the ‘plain’ semigroup context, in the inverse semigroup context (in the first two instances) and in the two-sided restriction semigroup context (in all but the last instance). The author has previously shown that in the last of these contexts, the behavior is pathological: ‘almost all’ finite restriction semigroups are inherently nonfinitely based. Here we show that this is not the case for left restriction semigroups, by exhibiting identities for the above varieties and for their joins with monoids (the analog of groups in this context). We do so by structural means involving subdirect decompositions into certain primitive semigroups. We also show that each identity has a simple structural interpretation.