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Ribeiro Soares Gonçalves de Araújo, João Jorge
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- Primitive groups synchronize non-uniform maps of extreme ranksPublication . Araújo, João; Cameron, Peter J.Let Ω be a set of cardinality n, G a permutation group on Ω, and f : Ω → Ω a map which is not a permutation. We say that G synchronizes f if the semigroup hG, fi contains a constant map.The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree n primitive groups synchronize maps of rank n − 1 (thus, maps with kernel type (2, 1, . . . , 1)). We prove some extensions of Rystsov’s result,including this: a primitive group synchronizes every map whose kernel type is (k, 1, . . . , 1). Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and n − 2. These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undirected) graph. The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.
- Computer solutions of problems in inverse semigroupsPublication . Araújo, João; McCune, WilliamIn 1981, Tamura posed a number of problems regarding the axiomatic definition of inverse semigroups. The main goal of this article is to use automated reasoning to solve these problems. In the process, we find some new defining sets of identities for the class of inverse semigroups and provide a single identity for groups in terms of two binary operations.
- Automorphisms of endomorphism monoids of relatively free bandsPublication . Araújo, João; Konieczny, JanuszFor a set X and a variety V of bands, let BV(X) be the relatively free band in V on X. For an arbitrary band variety V and an arbitrary set X, we determine the group of automorphisms of End(BV(X)), the monoid of endomorphisms of BV(X).
- Personal Learning Environments (PLE) in a distance learning course on mathematics applied to businessPublication . Bidarra, José; Araújo, JoãoThis paper argues that the dominant form of distance learning that is common in most e-learning systems rests on a set of learning devices and environments that may be outdated from the student’s perspective, namely because it is not supportive of learner empowerment and does not facilitate the efforts of self-directed learners. For this study we gathered and examined data on student’s use of Personal Learning Environments (PLEs) within a course on Mathematics Applied to Business offered by the Portuguese Open University (Universidade Aberta). We base the discussion on aspects that characterize student’s conceptions of PLEs, the emergence of connectivism as a new account of how learning occurs in a networked global environment, and conclude that an important goal of online course design should be to let students explore what the emergent Web 2.0 tools have to offer in distance learning. The widespread adoption of PLEs, bringing together learning from different contexts and sources of learning, shows that students are capable of expression in different forms, generating an added-value to distance learning environments.
- On finite complete presentations and exact decompositions of semigroupsPublication . Araújo, João; Malheiro, AntónioWe prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation. It is also proved that a semigroup M 0[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.
- The existential transversal property: a generalization of homogeneity and its impact on semigroupsPublication . Araújo, João; Bentz, Wolfram; Cameron, PeterLet G be a permutation group of degree n, and k a positive integer with k ≤ n. We say that G has the k-existential transversal property, or k-et, if there exists a k-subset A (of the domain Ω) whose orbit un- der G contains transversals for all k-partitions P of Ω. This property is a substantial weakening of the k-universal transversal property, or k-ut, investigated by the first and third author, which required this condition to hold for all k-subsets A of the domain Ω. Our first task in this paper is to investigate the k-et property and to decide which groups satisfy it. For example, it is known that for k < 6 there are several families of k-transitive groups, but for k ≥ 6 the only ones are alternating or symmetric groups; here we show that in the k-et context the threshold is 8, that is, for 8 ≤ k ≤ n/2, the only transitive groups with k-et are the symmetric and alternating groups; this is best possible since the Mathieu group M24 (degree 24) has 7-et. We determine all groups with k-et for 4 ≤ k ≤ n/2, up to some unresolved cases for k = 4, 5, and describe the property for k = 2, 3 in permutation group language. These considerations essentially answer Problem 5 proposed in the paper on k-ut referred to above; we also slightly improve the classification of groups possessing the k-ut property. In that earlier paper, the results were applied to semigroups, in particular, to the question of when the semigroup 〈G, t〉 is regular, where t is a map of rank k (with k < n/2); this turned out to be equivalent to the k-ut property. The question investigated here is when there is a k-subset A of the domain such that 〈G, t〉 is regular for all maps t with image A. This turns out to be much more delicate; the k-et property (with A as witnessing set) is a necessary condition, and the combination of k-et and (k − 1)-ut is sufficient, but the truth lies somewhere between. Given the knowledge that a group under consideration has the necessary condition of k-et, the regularity question for k ≤ n/2 is solved except for one sporadic group. The paper ends with a number of problems on combinatorics, permutation groups and transformation semigroups, and their linear analogues.
- A method for finding new sets of axioms for classes of semigroupsPublication . Araújo, João; Konieczny, JanuszWe introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.
- Semigroups of transformations preserving an equivalence relation and a cross-sectionPublication . Araújo, João; Konieczny, JanuszFor a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ,R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ,R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T (X, ρ,R) in terms of Green’s relations, describe the regular elements of T (X, ρ,R), and determine the following classes of the semigroups T (X, ρ,R): regular, abundant, inverse, and completely regular.
- Primitive permutation groups and strongly factorizable transformation semigroupsPublication . Araújo, João; Bentz, Wolfram; Cameron, PeterLet Ω be a finite set and T (Ω) be the full transformation monoid on Ω. The rank of a transformation t ∈ T (Ω) is the natural number |Ωt|. Given A ⊆ T (Ω), denote by 〈A〉 the semigroup generated by A. Let k be a fixed natural number such that 2 ≤ k ≤ |Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformations t ∈ T (Ω), every element in St := 〈G, t〉 can be written as a product eg, where e2 = e ∈ St and g ∈ G. In the second part we prove, among other results, that if S ≤ T (Ω) and G is the normalizer of S in the symmetric group on Ω, then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all s ∈ S there exists s′ ∈ S such that s = ss′s.) The paper ends with a list of problems.
- The commuting graph of the symmetric inverse semigroupPublication . Araújo, João; Bentz, Wolfram; Konieczny, JanuszThe commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dol˘zan and Oblak claimed that this upper bound is in fact the exact value.The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.