One-sided congruences on I_n, T_n and related semigroups
Lead Research Organisation:
University of York
Department Name: Mathematics
Abstract
This is a research project in semigroup theory, which is a branch of algebra.
Recall that I_n is the inverse monoid of partial bijections of n where n={1,2,...,n}. Every finite inverse monoid embeds into some I_n. Similarly, T_n is the monoid of all maps of n and every finite monoid embeds into some T_n.
The ideals of I_n and T_n are well known and easy to describe in terms of the ranks of elements, where the rank of a map f and is the cardinality of the image of f.
Ideals give certain congruences called Rees congruences. All congruences on
I_n and T_n are known. For the former, there is a machinery to describe congruences on inverse semigroups using `trace' and `kernel' and for the latter it has been calculated explicitly, see [2,Chapter 6]. In both cases, the congruences have a nice form. For example for any congruence on T_n there exists a k in n such that I_{k-1} (the ideal of all maps of rank no greater than k-1) is a congruence class, the classes in T_n\I_k are singletons, and everything interesting happens in I_k.
Surprisingly, one-sided congruences on I_n and T_n are little understood.
The aim of the research proposal is to
1. Describe the lattice of left congruences on I_n.
Due to the involution on any inverse semigroup, this will also be isomorphic to the lattice of right congruences.
2. Describe the lattice of left congruences on T_n and the lattice of right congruences on T_n.
It is well known that the wreath product GwrT_n is isomorphic to the endomorphism monoid of a free G-act with n generators, and the latter is important because it is an example of an independence algebra. Concretely, GwrT_n =GxGx...xGxT_n (n copies of G) with multiplication
(g_1,..., g_n,a)(h_1,...h_n,b)=(g_1h_1a, ..., g_nh_na,ab).
3. Describe the one-sided congruences on GwrT_n in terms of the one-sided congruences on T_n.
Motivation: (i) Knowing the one-sided congruences exactly determines the monogenic acts (the analogue of a module over a ring). (ii) Araujo, Bentz and Gomes [1] are looking at congruences on T_nxT_n. Ruskuc and co-authors are classifying congruences on partition monoids (which contain transformation monoids) [4]. In view of the recent interest of his team, Ruskuc and others, it seems the theory of (one-sided) congruences is currently very popular.
[1] J. Araujo, W. Bentz and G. Gomes, `Congruences on Direct Products of Transformation and Matrix Monoids', https://arxiv.org/abs/1602.06339.
[2] O. Ganyushkin and W. Mazorchuk, Classical finite transformation semigroups, Algebra and Applications Vol. 9, Springer 2009.
[3] V. Gould, `Independence algebras', Algebra Universalis 33 (1995), 294-318.
[4] J. East, J. Mitchell N. Ruskuc and M. Torpey, `Congruence lattices of finite diagram monoids', https://arxiv.org/abs/1709.00142.
Recall that I_n is the inverse monoid of partial bijections of n where n={1,2,...,n}. Every finite inverse monoid embeds into some I_n. Similarly, T_n is the monoid of all maps of n and every finite monoid embeds into some T_n.
The ideals of I_n and T_n are well known and easy to describe in terms of the ranks of elements, where the rank of a map f and is the cardinality of the image of f.
Ideals give certain congruences called Rees congruences. All congruences on
I_n and T_n are known. For the former, there is a machinery to describe congruences on inverse semigroups using `trace' and `kernel' and for the latter it has been calculated explicitly, see [2,Chapter 6]. In both cases, the congruences have a nice form. For example for any congruence on T_n there exists a k in n such that I_{k-1} (the ideal of all maps of rank no greater than k-1) is a congruence class, the classes in T_n\I_k are singletons, and everything interesting happens in I_k.
Surprisingly, one-sided congruences on I_n and T_n are little understood.
The aim of the research proposal is to
1. Describe the lattice of left congruences on I_n.
Due to the involution on any inverse semigroup, this will also be isomorphic to the lattice of right congruences.
2. Describe the lattice of left congruences on T_n and the lattice of right congruences on T_n.
It is well known that the wreath product GwrT_n is isomorphic to the endomorphism monoid of a free G-act with n generators, and the latter is important because it is an example of an independence algebra. Concretely, GwrT_n =GxGx...xGxT_n (n copies of G) with multiplication
(g_1,..., g_n,a)(h_1,...h_n,b)=(g_1h_1a, ..., g_nh_na,ab).
3. Describe the one-sided congruences on GwrT_n in terms of the one-sided congruences on T_n.
Motivation: (i) Knowing the one-sided congruences exactly determines the monogenic acts (the analogue of a module over a ring). (ii) Araujo, Bentz and Gomes [1] are looking at congruences on T_nxT_n. Ruskuc and co-authors are classifying congruences on partition monoids (which contain transformation monoids) [4]. In view of the recent interest of his team, Ruskuc and others, it seems the theory of (one-sided) congruences is currently very popular.
[1] J. Araujo, W. Bentz and G. Gomes, `Congruences on Direct Products of Transformation and Matrix Monoids', https://arxiv.org/abs/1602.06339.
[2] O. Ganyushkin and W. Mazorchuk, Classical finite transformation semigroups, Algebra and Applications Vol. 9, Springer 2009.
[3] V. Gould, `Independence algebras', Algebra Universalis 33 (1995), 294-318.
[4] J. East, J. Mitchell N. Ruskuc and M. Torpey, `Congruence lattices of finite diagram monoids', https://arxiv.org/abs/1709.00142.
Organisations
People |
ORCID iD |
| Victoria Gould (Primary Supervisor) | |
| Matthew Brookes (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509802/1 | 01/10/2016 | 31/03/2022 | |||
| 1941995 | Studentship | EP/N509802/1 | 01/10/2017 | 30/09/2019 | Matthew Brookes |
| Description | A left congruence P on a semigroup S is an equivalence relation that is compatible with the multiplicative structure of S (if a P b and c is an element of S then ca P cb). A two sided congruence is an equivalence relation that is both a left congruence and a right congruence. Congruences on semigroups are the analogue of normal subgroups in group theory and ideals in ring theory, determining homomorphic images. The analogue for a semigroup S of a module over a ring is a semigroup action, or S-act, which is defined just as a for group action. In ring theory cyclic modules are exactly determined by one sided ideals, monogenic S-acts are correspondingly determined by left congruences. Given a monogenic action of S (an S-act A such that there is an element b of A with A={sx | s an element of S}) the equivalence relation induced on S is a left congruence. A semigroup is inverse if for each element a of S there is a unique a^{-1} in S such that a(a^{-1})a=a and (a^{-1})a(a^{-1})=a^{-1}. Natural examples of inverse semigroups include groups and semilattices (a commutative semigroup in which every element is an idempotent). In fact, the set of idempotents {e | e^2=e}, is an important substructure of any inverse semigroup; it is a semilattice. It is well known that one and two sided congruences on inverse semigroups are determined by the trace (the restriction of the (one sided) congruence to the idempotents) and the kernel (the union of equivalence classes that contain idempotents). The trace is a congruence on the semilattice of idempotents and the kernel is a full subsemigroup. For one sided congruences this kernel-trace description has certain drawbacks including that the kernel map (the function taking a left congruence to its kernel) is neither join nor intersection preserving, and it is hard to determine whether a given subsemigroup is the kernel of a left congruence. I have refined the description of one sided congruences on inverse semigroups and shown that they are determined by the trace and a full inverse subsemigroup, termed the inverse-kernel. This is a substantially more powerful description of the lattice of one sided congruences, for instance the inverse-kernel map (the function taking a left congruence to its inverse kernel) preserves intersection and the set of inverse kernels is exactly the set of full inverse subsemigroups. It then follows that the set of left congruences (regraded as a semilattice) is a subdirect product of the semilattice of congruences on the idempotents and the semilattice of full inverse subsemigroups. The lattices of left and right congruences on an inverse semigroup are naturally isomorphic via the isomorphism defined by P \mapsto P_{-1} = { (a^{-1}, b^{-1}) | (a, b) \in P}. The inverse-kernel description for one sided congruences is intrinsically linked to this isomorphism, a pair (P,T) is the trace and inverse-kernel of a left congruence if and only if it is the trace and inverse-kernel of a right congruence and the corresponding right congruence is P_{-1}. The symmetric inverse monoid I_n is the set of partial bijections of the set {1,2,...,n}, under composition as binary relations. The symmetric inverse monoid is of particular importance within the class of inverse semigroups; it plays the role held by the symmetric group within group theory in that every inverse semigroup embeds into some symmetric inverse monoid (The Vagner-Preston representation theorem). Utilising the inverse-kernel approach to one sided congruences I have described the lattice of one sided congruences on I_n in terms of full inverse semigroups of I_n and congruences on the idempotents of I_n. A natural generalisation of I_n comes from the field of universal algebra. We consider the set {1,...,n} as an independence algebra (a family of algebras which have a well defined notion of a basis), then I_n is the partial automorphism monoid. Other examples of independence algebras include vector spaces and free group actions (free G-acts). It is well known that the partial automorphism of a free G-act with n generators may be realised as a partial wreath product GwrI_n. I have provided a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. I have further described the constituent parts in terms of subgroups of direct and semidirect products of groups. This description is utilised to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and on the number of generators of the action. I have also described one sided congruences on GwrI_n in terms of one sided congruences on I_n and one sided congruences on the Clifford semigroup (G^0)^n. |
| Exploitation Route | One sided congruences on semigroups are closely related to several properties of semigroups. For instance a semigroup is right Noetherian if all the right congruences are finitely generated, and a semigroup is right coherent if for any finitely generated right congruence P on S and any elements a,b in S both the subact (aP)S \cap (bP)S of the right S-act S/P is finitely generated and the annihilator r(aP)={(u,v) | au P av} is finitely generated. Thus the inverse-kernel description for one sided congruences may be used to determine when inverse semigroups satisfy these properties. |
| Sectors | Other |
| URL | https://arxiv.org/abs/1901.11455 |