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Joao Araujo

Publications and source records attributed to Joao Araujo.

15 recordsLinked to original sources
arXiv2026

Resilient AI Supercomputer Networking using MRC and SRv6

Tail latency dominates the performance of synchronous pretraining jobs when running at very large scales. We describe a three-pronged approach: (1) a new RDMA-based transport protocol, MRC, sprays across many paths and actively load-balances between them, eliminating the issue of flow collisions (2) the use of multi-plane Clos topologies to get the benefits of high switch radix and redundancy, allowing training clusters well over 100K GPUs to be built as two-tier topologies while increasing physical redundancy, and (3) the use of static source-routing using SRv6 to allow MRC the freedom to bypass failures by itself. We describe our experiences running MRC and static SRv6 routing in production in OpenAI and Microsoft's largest training clusters, where it has been used to train the latest frontier models. We demonstrate how MRC allows AI training jobs to ride out many network failures that previously would have interrupted training.

cs.NI↗
arXiv2018

Conjugacy in inverse semigroups

In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a\sim_{\mathrm{i}} b$, if there exists $g\in S^1$ such that $g^{-1} ag=b$ and $gbg^{-1} =a$. The purpose of this paper is to study the conjugacy $\sim_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister $P$-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.

math.GR↗
arXiv2014

The rank of the semigroup of transformations stabilising a partition of a finite set

Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a full transformation $f:X\to X$ preserves (or stabilizes) the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $\mathcal{P}$. In 2005 Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture. The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem. The paper ends with a number of problems for experts in group and semigroup theories.

math.GR↗
arXiv2013

Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

math.GR↗
arXiv2012

A 2-base for inverse semigroups

An open problem in the theory of inverse semigroups was whether the variety of such semigroups, when viewed as algebras with a binary operation and a unary operation, is 2-based, that is, has a base for its identities consisting of 2 independent axioms. In this note, we announce the affirmative solution to this problem: the identities \[ \quad x(x'x) = x \qquad \quad x (x' (y (y' ((z u)' w')'))) = y (y' (x (x' ((w z) u)))) \] form a base for inverse semigroups where ${}'$ turns out to be the natural inverse operation. We recount here the history of the problem including our previous efforts to find a 2-base using automated deduction and the method that finally worked. We describe our efforts to simplify the proof using \textsc{Prover9}, present the simplified proof itself and conclude with some open problems.

math.GR↗
arXiv2011

A characterization of adequate semigroups by forbidden subsemigroups

A semigroup is \emph{amiable} if there is exactly one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups which do not contain isomorphic copies of two particular nonadequate semigroups as subsemigroups.

math.GR↗
arXiv2011

On a problem of M. Kambites regarding abundant semigroups

A semigroup is \emph{regular} if it contains at least one idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class. A regular semigroup is \emph{inverse} if satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class, or (ii) the idempotents commute. Analogously, a semigroup is \emph{abundant} if it contains at least one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. An abundant semigroup is \emph{adequate} if its idempotents commute. In adequate semigroups, there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class, must the idempotents commute? In this note we use ideal extensions to provide a negative answer to this question.

math.GR↗
arXiv2011

Independent axiom systems for nearlattices

A nearlattice is a join semilattice such that every principal filter is a lattice with respect to the induced order. Hickman and later Chajda et al independently showed that nearlattices can be treated as varieties of algebras with a ternary operation satisfying certain axioms. Our main result is that the variety of nearlattices is 2-based, and we exhibit an explicit axiom system of two independent identities. We also show that the original axiom systems of Hickman and of Chajda et al are, respectively, dependent.

math.CO↗
arXiv2010

Minimal paths in the commuting graphs of semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\geq4$, then, with a few exceptions, the diameter of $\cg(J)$ is 5. On the other hand, we prove that for every positive integer $n$, there exists a semigroup $S$ such that the diameter of $\cg(S)$ is $n$. We also study the left paths in $\cg(S)$, that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all $i\in \{1,\ldot, m\}$. We prove that for every positive integer $n\geq2$, except $n=3$, there exists a semigroup whose shortest left path has length $n$. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.

math.GR↗
arXiv2010

An elegant 3-basis for inverse semigroups

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: [\quad x=(xx')x \qquad \quad (xx')(y'y)=(y'y)(xx') \qquad \quad (xy)z=x(yz"). ] The goal of this note is to prove the converse, that is, we prove that an algebra of type $<2,1>$ satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.

math.GR↗
arXiv2010

Axioms for unary semigroups via division operations

When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The paper closes with a list of open problems.

math.GR↗
arXiv2009

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also classify the permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G, a}\setminus G$ and $\genset{a^g | g\in G}$ are generated by their idempotents for all non-invertible transformations of $X$.

math.GR↗
arXiv2008

Embedding properties of endomorphism semigroups

Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, and by Sub(V) (resp., End(V)) the collection of all subspaces (resp., endomorphisms) of a vector space V. We prove various results that imply the following: (1) If X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2^card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),\cap) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2^card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2^card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, of left ideals). Then End(F) has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras endowed with a notion of independence (in particular independence algebras).

math.RA↗
arXiv2008

The Rank of the Endomorphism Monoid of a Partition

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.

math.GR↗