Title: The Precedence-Constrained Family Traveling Salesman Problem

Speaker:  Raquel Bernardino, Department of Mathematics, Lisbon School of Economics and Management (ISEG)

Date | Time: April 8, 2026 | 14h30

Place: NOVA FCT,

Abstract:

We introduce the Precedence-Constrained Family Traveling Salesman Problem (PC- FTSP), which generalizes the FTSP by adding precedence relations between families. The PC-FTSP arises in warehouse order picking with scattered storage, where heavier items must be picked before lighter ones. We propose several formulations and compare them theoretically and empirically. We also derive valid inequalities that strengthen the linear programming (LP) relaxations. As some formulations are non- compact, we design a branch-and-cut algorithm to solve them. Results show the non- compact formulations outperform the compact ones, and the inequalities effectively strengthen the LP bounds on our instances.

Room 1.12 Building VII
   
Abstract: In this talk we will address the lifting problem for the Černý conjecture: namely, whether the validity of the conjecture for a quotient automaton can always be transferred (or “lifted”) to the original automaton. Although a complete solution remains open, we show that it is sufficient to verify the Černý conjecture for three specific subclasses of reset automata: radical, simple, and quasi-simple.  Our approach relies on establishing a Galois connection between the lattices of congruences and ideals of the transition monoid. This connection not only serves as the main tool in our proofs but also provides a systematic method for computing the radical ideal and for deriving structural insights about these classes.  
This is a joint work with Prof. Emanuele Rodaro.

Wave scattering is present in many applications regarding non destructive testing, namely medical imaging. In particular, in the context of inverse scattering problems, the goal is to find unknown properties of the obstacle (scatterer) from the knowledge of the incident field and the measurement of the scattered field or far-field pattern. In this talk, we will focus on numerical methods to solve mathematical models for time-harmonic scattering problems. We will focus on new approaches to ensure stability of the method of fundamental solutions to solve the direct problem and in numerical methods that deal with the ill-conditioning and nonlinearity of the inverse problem, in a time-harmonic wave propagation context. Finally, we will consider a numerical approach for a toy model for elastography, a medical imaging modality used for biological tissue imaging, focusing on their applications in the retina.

Room 1.12 ed. VII 
 
Abstract:  A minimal generating set of a group is a set of elements that generates the group such that, if any element is removed, the remaining set no longer generates the group. For example, the symmetric group of degree n has minimal generating sets whose sizes range from 2 up to n−1. The study of minimal generating sets is a topic of particular interest because of its connections with two important areas: discrete geometry and computational algebra. In discrete geometry, there is a well-known one-to-one correspondence between certain minimal generating sets of involutions satisfying specific axioms and regular polytopes. In computational algebra, the connectivity of product replacement graphs, an important tool used in algebra systems to generate random elements of groups, depends on the parameters m(G) and d(G), which denote the maximal and minimal sizes of minimal generating sets of a group G, respectively.

In this seminar, I will give an overview of the topic and discuss some of its main results and applications.

This talk presents a numerical study of a system of partial differential equations consisting of a parabolic, an elliptic, and a hyperbolic equation, that be used to describe cell migration in elastic tissues by integrating chemotaxis, interstitial fluid pressure, and tissue displacement. In this framework, the system couples a Keller–Segel-type model for cell density and chemical concentration with an elliptic equation for fluid pressure and a wave-type equation for tissue displacement. Cell transport is driven by a nonlinear convective velocity that depends on the chemoattractant gradient, Darcy flow induced by the pressure field, and the tissue displacement velocity, while chemoattractant transport is advected only by Darcy flow. We propose a semidiscrete finite difference method on nonuniform grids, which can be seen as a fully discrete-in-space piecewise linear finite element method. Using an appropriate discrete norm, we establish second-order spatial error estimates for all variables. Finally, we present numerical simulations illustrating the theoretical results and the interplay between chemotaxis and tissue mechanics.

Statistics and Risk Management Seminar

Department of Mathematics, NOVA MATH/FCT NOVA

Title: A computational tool for unsupervised variable selection and patient stratification

Speaker: Roberta Coletti, NOVA Math

Date | Time: May 04, 2026 | 12:00

Location: VII-1.11

Teams: https://teams.microsoft.com/meet/36124291051668?p=9wzuxJVsdpJQkGb9ww

Abstract: In this study, we developed TRIM-IT, a computational framework for data-driven patient stratification and biomarker discovery based on high-dimensional omics data. To address the challenges associated with high dimensionality, TRIM-IT first performs unsupervised variable selection to reduce data complexity while preserving dataset structure. The selected variables are then used for unsupervised clustering and downstream analyses to characterize the identified patient groups. Applied to glioblastoma transcriptomics data, TRIM-IT uncovered three distinct patient clusters associated with tumor histology, significantly different survival outcomes, and molecular profiles suggestive of potential biomarker candidates. 

Short Bio: Roberta Coletti studied mathematics at Sapienza University of Rome. She completed a PhD in mathematics at the University of Trento, with a thesis on ordinary differential equation models of prostate cancer immunotherapy. Between 2021 and 2024, she was a researcher at the Center for Mathematics and Applications at NOVA University of Lisbon. Her work focuses on identifying molecular biomarkers for glioma cancer by analyzing large multi-omics datasets using statistical and machine learning methods. 

Organizers: Isabel Natário and Mina Norouzirad and Marta Lopes

LogosTodos.JPG

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (https://doi.org/10.54499/UID/00297/2025) and UID/PRR/00297/2025 (https://doi.org/10.54499/UID/PRR/00297/2025) (Center for Mathematics and Applications - NOVA Math)

Abstract:
Shigesada, Kawasaki, and Teramoto showed that introducing cross-diffusion effects in the spatial Lotka–Volterra competition
model can destabilise the spatially homogeneous equilibrium and generate spatial patterns. In this seminar, I will present recent work in collaboration with Tommaso Lorenzi and Gaetana Gambino, in which we examined the effect of phenotype heterogeneity in competing populations. In particular, in our modelling framework, phenotype diversity affects movement and interactions between individuals.
Motivated by cellular plasticity, we considered a regime of rapid phenotype switching and derived conditions for cross-diffusion- and phenotype-driven instabilities. Finally, I will present numerical simulations to illustrate the role of the phenotype distribution and its impact on competitive outcomes.

This Seminar is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P.,  under the scope of the projects \href{https://doi.org/10.54499/UID/00297/2025}{\textcolor{blue}{UID/00297/2025}} and \href{https://doi.org/10.54499/UID/PRR/00297/2025}{\textcolor{blue}{UID/PRR/00297/2025}} (Center for Mathematics and Applications

Title: Nearest Correlation Matrices via Halfspace Projections  
Speaker: Yunier Bello-Cruz, Department of Mathematical Sciences at Northern Illinois University (NIU), IL, USA
Date | Time: May 6, 2026 | 14h30 
 

Abstract:

The nearestcorrelation matrix problem asks for the closest positive semidefinite matrixwith unit diagonal to a given symmetric matrix G, measured in the Frobeniusnorm. We introduce HBAP (Halfspace Best-Approximation Projection), aprojection-based algorithm that approximates the positive semidefinite cone byintersections of supporting halfspaces and computes iterates via projectionsonto these simpler sets. At each iteration, HBAP constructs two halfspaces, asupporting halfspace for the positive semidefinite cone derived from thesquared-distance function, and a localization halfspace that enforces monotoneprogress, and projects the anchor point onto their intersection with theunit-diagonal affine subspace. Every iterate satisfies the unit-diagonal constraintby construction. We prove that HBAP is well defined and that the full sequenceconverges to the nearest correlation matrix to G. The projection subproblem ateach step admits a closed-form reduction to a 2X2 linear complementarityproblem, yielding an explicit update formula. Alternatively, the sameprojection can be computed via Dykstra-type inner iterations, for which wederive a natural splitting with fully explicit projectors.

Room: to be announced
 
Abstract: Algorithms and programs for computing the k-universal transversal property of a permutation group G are described. This property gives information about the regularity of a transformation semigroup <G,t> generated by G and a transformation t. New results obtained by the programs are also described.

In this seminar, we present sufficient conditions for the convergence of asymptotic systems in non-autonomous Cohen-Grossberg neural network models that incorporate both infinite discrete time-varying and distributed delays. The main stability criterion is obtained by imposing conditions under which the non-delay terms asymptotically dominate the delay terms. As an application, we provide sufficient conditions ensuring that all solutions of a non-periodic neural network model with unbounded delays converge to a periodic function as time goes to infinity. A numerical example is presented to illustrate the effectiveness of the new results. This is a joint work with A. Elmwafy (PhD student) and César M. Silva of University of Beira Interior, Portugal.

Abstract:

Several strategies are described and analyzed to speed-up gradient-type methods when applied to the minimization of strictly convex quadratics and strictly convex functions. The proposed techniques focus on relaxing the traditional optimal step length associated with gradient methods, including the steepest descent (SD) and the minimal residual (MR) methods. Such a relaxation avoids the well-known negative zigzag effect and allows the iterates to move in the entire space which in turn implies that every so often the search direction approaches some eigenvector of the underlying Hessian matrix. The proposed speedups then rely on taking advantage of the properties of the Lanczos method once a search direction that approaches an eigenvector has been identified in order to accelerate the convergence towards the global minimizer. After analyzing the proposed strategies, we illustrate them on the global minimization of strictly convex functions.

How do we control the sublinear Hardy–Littlewood maximal operator in non-Euclidean settings?  By trading it for linear alternatives.  In this talk, we show that the operator's boundedness on variable Lebesgue spaces over spaces of homogeneous type is entirely characterized by the uniform boundedness of specific averaging operators over dyadic cubes.  Adapting Lars Diening's 2005 Euclidean framework, we expose the mechanics behind this clean reduction: Musielak–Orlicz spaces, Hytönen–Kairema adjacent grids, and the omnipresent principle of self-improvement.

Abstract:

In this work, we study two-stage distributionally robust mixed-integer programs with uncertain parameters having finite discrete support. This setting is particularly attractive from a computational perspective because the DRO problem remains tractable and can be reformulated as a (possibly much larger) mixed-integer program, allowing the use of standard optimization technology. We propose an ambiguity set defined through a generalized optimal transportation problem. This formulation extends the classical Kantorovich ambiguity set, enabling the modeling of a wider range of practical situations while preserving useful structural properties. We analyze the main characteristics of this ambiguity set and discuss how they affect the structure of the resulting optimization problem.

Three solution frameworks are investigated. The first is the Benders-like method proposed by Bansal, Huang, and Mehrotra (2018). The second derives a single-stage formulation obtained by dualizing the transportation problem defining the ambiguity set. The third uses an epigraph formulation with dynamically generated optimality cuts in a row-and-column generation scheme. The approaches are evaluated on a distributionally robust location–transportation problem with uncertain demand. Computational results show that the relative performance of the approaches depends strongly on the characteristics of the ambiguity set: the dualization approach is simple and effective in the easiest instances, the row-and-column generation performs well when the ambiguity set approximates robust optimization, and the Benders-type method is preferable in the remaining settings.

This talk presents  a general theory for coupled systems, with regular and singular nonlinear fully differential equations of higher order, with generalized impulsive effects, depending on both variables and some derivatives. Two types of results are shown: first, an existence theorem, proved via fixed point theory. Secondly, we define a new type of coupled lower and upper solutions, and sufficient conditions to get the localization of a solution and some of its derivatives. The method is based on considering an auxiliary, truncated, and perturbated problem, whose solutions are also solutions of the initial problem. These results are applied to systems of singular Laplacians.

Abstract: Let G be a simple graph and k a positive integer. Two important graph operations are the k-th graph power of G and the complement graph of G. We explore a permutation between both operations and how it affects the independence number of the resultant graphs.

We will discuss local aspects of Banach space theory arising in finite-dimensional Banach spaces of analytic polynomials. The lecture will focus on how classical constants, such as projection constants, unconditional basis constants, and Gordon-Lewis constants, reflect the geometry of these polynomial spaces. I will explain the main ideas behind recent estimates and indicate how they connect local Banach space theory with complex analysis, probability, and combinatorial methods. The lecture is based on joint works with A. Defant, D. Galicer, M. Mansilla, and S. Muro.

Please see the attached PDF for the abstract.

https://drive.google.com/file/d/1V1smOS2b67Hm6UyL2hMkJRqDLSXhL7BH/view?usp=sharing

Abstract:  The identity checking problem asks whether an identity holds in a given semigroup. In this talk, we discuss the identity checking problem for diagram monoids, focusing on twisted partition monoids. We present a structural approach based on an embedding of the twisted partition monoid into a larger regular monoid satisfying the same identities. Using this approach, we show that for every n ≥ 5, the identity checking problem in the twisted partition monoid (P_n^\tau) is co-NP-complete.

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

Speaker: Weining Wang, University of Bristol

Abstract:

This two-lecture module introduces students to time series analysis and its modern extensions using deep learning and generative AI. The first lecture covers classical stochastic models (AR, MA, ARIMA) and fundamental concepts such as stationarity, forecasting, and model identification. The second lecture bridges traditional time series methods with modern AI approaches, focusing on Transformer architectures for sequence modeling and Diffusion models for generative time series tasks. The module emphasizes conceptual understanding, methodological connections, and practical relevance in economics and data science.

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

Speaker: Weining Wang, University of Bristol

Abstract:

This two-lecture module introduces students to time series analysis and its modern extensions using deep learning and generative AI. The first lecture covers classical stochastic models (AR, MA, ARIMA) and fundamental concepts such as stationarity, forecasting, and model identification. The second lecture bridges traditional time series methods with modern AI approaches, focusing on Transformer architectures for sequence modeling and Diffusion models for generative time series tasks. The module emphasizes conceptual understanding, methodological connections, and practical relevance in economics and data science.

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

https://eventos.fct.unl.pt/novamath_thematic_weeks/pages/seminars

Statistics and Risk Management Seminar (double session)

Department of Mathematics, NOVA MATH/FCT NOVA

Seminar I

Title: Induced nonparametric ROC surface regression

Speaker: Vanda Inácio; University of Edinburgh, Scotland

Date | Time: July 14, 2026 | 14:00

Location: Hangar II, room 2

Teams: https://teams.microsoft.com/meet/36124291051668?p=9wzuxJVsdpJQkGb9ww

Abstract: The receiver operating characteristic (ROC) surface is a popular tool for evaluating the discriminatory ability of diagnostic tests measured on a continuous scale when there exist three ordered disease groups. Motivated by the impact that covariates may have on the diagnostic accuracy, and to safeguard against model misspecification, we develop a flexible model for conducting inference about the covariate-specific ROC surface and its functionals. Specifically, we postulate a location-scale regression model for the test outcomes in each of the three disease groups where the mean and variance functions are estimated through penalised-splines, while the distribution of the error term is estimated via a smoothed version of the empirical cumulative distribution function of the standardised residuals. Our simulation study shows that our approach successfully recovers the true covariate-specific volume under the surface and optimal pair of thresholds in a variety of scenarios. Our methods are motivated by and applied to data derived from an Alzheimer's disease study and we seek to assess the accuracy of several  biomarkers to distinguish between individuals with normal cognition, mild cognitive impairment, and dementia and how this discriminatory ability may change with the age and gender.

Seminar II

Title: Extremal Vulnerability

Speaker: Miguel de Carvalho; University of Edinburgh & University of Aveiro

Date | Time: July 14, 2026 | 15:30

Location: Hangar II, room 2

Teams: https://teams.microsoft.com/meet/36124291051668?p=9wzuxJVsdpJQkGb9ww

Abstract: In many complex systems, identifying the most vulnerable component is essential for effective prevention, intervention, and risk management. In this talk, I will introduce the notion of extremal vulnerability, defined as the long run tendency of a component to be affected by extreme events occurring in other components. The proposed framework builds on the tail dependence matrix and introduces the Extremal Vulnerability Rank (XVRank) method—a PageRank-inspired algorithm designed to quantify extremal vulnerability.  We establish the theoretical properties of the proposed inferences, including consistency and asymptotic normality, and validate their performance through Monte Carlo simulations. The proposed methods are illustrated using financial data to determine assets most exposed to severe market downturns. Joint work with P. Redondo, R. Huser, and H. Ombao (KAUST).

Organizers: Isabel Natário and Mina Norouzirad and Miguel Fonseca

LogosTodos.JPG

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (\url{https://doi.org/10.54499/UID/00297/2025}) and UID/PRR/00297/2025 (\url{https://doi.org/10.54499/UID/PRR/00297/2025}) (Center for Mathematics and Applications)

Abstract:

Derivative-freeoptimization (DFO) is essential for problems where derivatives are unavailableor expensive to compute, but its scalability is limited by the high cost ofmodel construction in large dimensions. Subspace techniques address thischallenge by building models and performing optimization in low-dimensionalaffine subspaces.  This talk presents aunified view of subspace modelling and methods for DFO.  We first establish theoretical connectionsbetween full-space and subspace linear and quadratic models and simplexderivatives, showing that they coincide on the underlying subspace and alongorthogonal directions.  We then presentrandom subspace trust-region methods that leverage these models, includingframeworks with provable convergence and complexity guarantees forunconstrained and convex-constrained problems.

Title: Bivariate models involving independent gamma distributed components

Speaker: Barry C. Arnold, Department of Statistics, University of California, Riverside, USA

Date | Time: September 23, 2026 | 14:30

Location: Building I, Samsung room (Auditório Manuel Laranjeiro)  

Abstract: Several multivariate models involving independent gamma distributed components (three of which are new) are described. The flexible bivariate beta(2) model introduced by Arnold and Ng (2011) provides the template for the other models. It involved ratios of sums of independent gamma variables. The other models involve differences, sums, products and minima rather than ratios.

Short Bio: Barry C. Arnold is Distinguished Professor Emeritus at the Department of Statistics, University of California, Riverside, USA. He received his Ph.D. in Statistics from Stanford University in 1965. He has authored 14 books and more than 300 research papers in reputed peer-reviewed journals and contributed volumes. Professor Arnold has guided 17 Ph.D. scholars and has been on editorial boards of renowned journals. His research interests include estimation theory, probability, stochastic processes, mathematical learning models, biological models, characterizations, income distributions, order statistics, inequality measurement, record values, conditionally specified distributions, and Bayesian inference. He has been invited for scholarly lectures from across the world and is recognized by American Statistical Association, USA (Fellow); American Association for the Advancement of Science, USA (Fellow); Institute of Mathematical Statistics, USA (Fellow); Royal Statistical Society, UK (Fellow); International Statistical Institute, the Netherlands (Elected Member). He also received, in December 2023 an honorary doctorate from the Colegio de Postgraduados, Montecillo, Mexico.

Organizers: Isabel Natário and Mina Norouzirad and Carlos A. Coelho

LogosTodos.JPG

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (\url{https://doi.org/10.54499/UID/00297/2025}) and UID/PRR/00297/2025 (\url{https://doi.org/10.54499/UID/PRR/00297/2025}) (Center for Mathematics and Applications)

Course

Title: Beyond multivariate normality. A spectrum of alternative constructions of  bivariate and multivariate distributions

Speaker: Barry C. Arnold, Department of Statistics, University of California, Riverside, USA

Date | Time: September 24, 2026 | 10h-12h and 14h-16h

Location: Building VII, room 1.9  

Abstract: Part I: The classical multivariate normal distribution does constitute a good starting point for our discussion. Our goal is to document various methods for constructing more flexible families of multivariate distributions using well-known (or sometimes, little-known) models as components. We will give many examples of such augmented distributions, but we will not strive to provide an exhaustive listing of all examples that have appeared in the literature. The emphasis will be on the description of methods or algorithms of augmentation, with only a reasonably extensive sample of examples in which the techniques have been or could be used.Part II: In Chapter 2, we will review positive features of the classical multivariate normal model. However, partially to motivate the study of alternative multivariate models, we will next look carefully at the classical multivariate normal model, paying close attention to the ``baggage" that comes along with an assumption of the classical model as an acceptable descriptor of our data. If you accept classical multivariate normality, you must also accept an extensive list of items as being appropriate for your data modeling project.

Short Bio: Barry C. Arnold is Distinguished Professor Emeritus at the Department of Statistics, University of California, Riverside, USA. He received his Ph.D. in Statistics from Stanford University in 1965. He has authored 14 books and more than 300 research papers in reputed peer-reviewed journals and contributed volumes. Professor Arnold has guided 17 Ph.D. scholars and has been on editorial boards of renowned journals. His research interests include estimation theory, probability, stochastic processes, mathematical learning models, biological models, characterizations, income distributions, order statistics, inequality measurement, record values, conditionally specified distributions, and Bayesian inference. He has been invited for scholarly lectures from across the world and is recognized by American Statistical Association, USA (Fellow); American Association for the Advancement of Science, USA (Fellow); Institute of Mathematical Statistics, USA (Fellow); Royal Statistical Society, UK (Fellow); International Statistical Institute, the Netherlands (Elected Member). He also received, in December 2023 an honorary doctorate from the Colegio de Postgraduados, Montecillo, Mexico.

Organizers: Isabel Natário and Mina Norouzirad and Carlos A. Coelho

LogosTodos.JPG

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (\url{https://doi.org/10.54499/UID/00297/2025}) and UID/PRR/00297/2025 (\url{https://doi.org/10.54499/UID/PRR/00297/2025}) (Center for Mathematics and Applications)

Room:  1.19 do VII 
 
Abstract: The study and characterization of the subvarieties of W (variety of semigroups whose square is completely regular) when W is a E-semigroup (the set of its idempotents, E(S), is a subsemigroup[1]) has been proposed by Borralho and Kinyon[3] following a question from Araújo et al.[2].
Although in its formulation this problem had been limited to semigroups where the idempotents commute, called E-commutative semigroups[1], ∀e, f ∈ E(S), e.f = f.e, in this work, I will also present results for the E-semigroups in W. These can be defined by the additional condition ∀e, f ∈ E(S), f.e ∈ W(e.f), where W(a) represents the set of weak inverses of a, W(a) = {x : x.a.x = x}.
When idempotents form a subsemigroup, we call it a band. If they commute, E(S) is
a semilattice of idempotents; otherwise, it is a semilattice of rectangular bands. Although the former is a particular case of the latter, this presentation treats both situations distinctly, as they are different structures with different properties.

The point of view we adopt to study the structure of a relatively complex semigroup is to first decompose it into a collection of subsemigroups, each of which has a somewhat
simpler structure, and then compose a complicated and “bigger” semigroup from simpler
ones.
In this work, this is accomplished in 3 steps:
1. Dividing the semigroup S in disjunct closed classes Sα of known structure.
2. Defining a “gross” structure by studying the relation between these classes induced
by the semigroup operation, Sα.Sβ → Sγ.
3. Studying a “fine” structure that relates the positions of two elements from different
classes to the position of their product. For any S ∈ W and any a ∈ S, a2 lies in a group; thus S is an epigroup, also called group-bound in Howie[4]. Then S splits into unipotency classes, Ke, defined by elements whose square lies in Ge, the largest subgroup whose idempotent is e[5]. In W, these unipotency classes are disjoint subsemigroups of S and can be indexed by the elements of the semilattice E(S). Starting from S 2 and taking into account that E(S2) = E(S), we refer to the conditions for being able to define homomorphisms in S, φα,β : Sα → Sβ for all α ≥ β of the semilattice E(S), and so define a multiplication of the elements of the
different unipotency classes. This way, we can study the “fine” structure of S ∈ W when
their idempotents form a subsemigroup.
Except as stated otherwise, the basic concepts used here can be found in Howie[4] and
Shevrin[5].
References
[1] J. Almeida, J.-E. Pin, and P. Weil, Semigroups whose idempotents form a subsemigroup, Math.
Proc. Camb. Phi. Soc. 111 (1992), 241-253.
[2] J. Ara ́ujo, M Kinyon, J. Konieczny, and A. Malheiro, Four notions of conjugacy for abstract
semigroups, in: Proceedings of the Royal Society of Edinburgh Section A: Mathematics 147(6) (2017),
1169-1214.
[3] M. Borralho and M. Kinyon, Variants of epigroups and primary conjugacy, Commun. Algebra
48(12) (2020), 5465-5473.
[4] J.M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, New Series 12, (Oxford University Press, 1995).
[5] L.N. Shevrin, Epigroups, in Structural theory of automata, semigroups and universal algebra (Springer, 2005) 331-380.