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In this paper, we introduce the concepts of Bochner and Bohr almost automorphic functions on the semigroup induced by complete-closed time scales and their equivalence is proved. Particularly, when \begin{document}$ \Pi = \mathbb{R}^{+} $\end{document} (or \begin{document}$ \Pi = \mathbb{R}^{-} $\end{document} ), we can obtain the Bochner and Bohr almost automorphic functions on continuous semigroup, which is the new almost automorphic case on time scales compared with the literature [ 20 ] (W.A. Veech, Almost automorphic functions on groups, Am. J. Math., Vol. 87, No. 3 (1965), pp 719-751) since there may not exist inverse element in a semigroup. Moreover, when \begin{document}$ \Pi = h\mathbb{Z}^{+},\,h>0 $\end{document} (or \begin{document}$ \Pi = h\mathbb{Z}^{-},\,h>0 $\end{document} ), the corresponding automorphic functions on discrete semigroup can be obtained. Finally, we establish a theorem to guarantee the existence of Bochner (or Bohr) almost automorphic mild solutions of dynamic equations on semigroups induced by time scales.

This paper will discuss level of conceptual understanding of 18 mathematics students in learning elementary group theory during abstract algebra course 2016-2017 academic year at Andalas University. Participants were asked to answer three proof tests in relation to group theory. Students' solutions to the proof test were taken as the key source of data used to: (i) classify students to one of the four levels of conceptual understanding and (ii) analyze students errors in learning elementary group theory. One student for each level was interviewed to provide additional information about common students' errors on the proof task and to aid the process of understanding the underlying cause of these errors. The finding shows that: (1) Students' achievement in proof task is still problematic; (2) Most students have difficulties in verifying the existence of identity and inverse element; (3) Factors that contribute to errors in proof task are: lack of conceptual understanding and that student treated binary operations on a group as a binary operations on real numbers.

This paper proposes a new block encryption algorithm for cryptographic information protection. It describes a new transformation method EM (Exponentiation Module), which is part of the algorithm, and a method of S-box obtaining. The paper also considers an optimization technique to advance the efficiency of key selection and calculation. We discuss the possibility to obtain good results by applying the peculiar properties of cryptographic primitives in the Galois field. To increase the strength and speed of the encryption algorithm, we used a nonpositional polynomial notation and an indexed view for the Galois field. The paper provides for statistical properties of the ciphertext obtained with the developed algorithm. We also present the results of differential and linear cryptanalysis of the S-box used.

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P k (A). We show that P1(A) is a subset of P k (A) for all k. If T in P1(A), then Tn lies in P1(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P k (A), then UTV lies in P k (A) for all natural k. Let A be unital, then 1) if an element T in P1(A) is right invertible, then any right inverse element T−1 lies in P1(A); 2) for ssIss = 1 the class P1(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P1(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P1(A) coincides with the set of all paranormal operators on a Hilbert space H.

The inverse finite element method (iFEM) is a mechanics-based algorithm for deformed-shape estimation of structures. In this paper, an enhanced inverse beam element named iEBT2 is developed based on classical beam theory. The element formulation is derived by minimizing a weighted-errors functional that consists of experimental and numerical section strains. The improved coefficient matrix KR is always non-singular, assuring that the solution of iFEM formulation exists. Location-independent feature of matrix KR simplifies the inverse finite element modeling, especially for complicated structures. Weighting constants are utilized to define error functional, aiming to penalize the contributions from “measure-less” stations. Numerical and experimental cases have been performed and demonstrated excellent predictive capability when iEBT2 model has complete strain measures. In the case of missing strain components, iEBT2 enables deformation estimation and its accuracy is acceptable in general. The enhanced inverse element extends the practical usefulness of iFEM in shape-sensing analysis of civil infrastructures.

We characterize the elements with outer inverse in a semigroup S, and provide explicit expressions for the class of outer inverses b of an element a such that bS⊆yS and Sb⊆Sx, where x, y are any arbitrary elements of S. We apply this result to characterize pairs of outer inverses of given elements from an associative ring R, satisfying absorption laws extended for the outer inverses. We extend the result on right–left symmetry of aR⊕bR=(a+b)R (Jain–Prasad, 1998) to the general case of an associative ring. We conjecture that ‘given an outer inverse x of a regular element a in a semigroup S, there exists a reflexive generalized inverse y of a such that x≤−y’ and prove the conjecture when S is an associative ring.

For any semigroup S and any , the author [Linear Algebra Appl. 2012;436:1909–1923] introduced the idea of the (b, c)-inverse y of a, and then [Bicommuting properties of generalized inverses. Linear...

The classical Cayley-Hamilton theorem is extended to Drazin inverse matrices and to standard inverse matrices. It is shown that knowing the characteristic polynomial of the singular matrix or nonsingular matrix, it is possible to write the analog Cayley-Hamilton equations for Drazin inverse matrix and for standard inverse matrices.

In order to prepare our students – future teachers of mathematics for potential problems, their pupils may encounter in math classes when working with binary operations, such as addition, multiplication, etc., subjects Elementary Algebra and Mathematics and Its Teaching Methodology are included in the study plans. In these subjects, when working with unusual binary operations, students get into the role of the pupils and are forced to think about each next step in their investigation.

In this paper, we give new existence criteria for the inverse along an element by means of Drazin inverses, one-sided annihilator ideals, idempotents and one-sided invertibility of certain elements in semigroups and rings, respectively. In addition, we take advantage of idempotents and one-sided principal ideals to characterize the inner inverse along an element in a ring. Furthermore, their expressions are shown. Next, we obtain Cline’s formula for the inverse along an element. Finally, we study the inverse of the product along an element in a semigroup. In particular, some results in this paper recover the consequences of the classical generalized inverses such as group inverse, Drazin inverse, and Moore-Penrose inverse.

A semigroup S is called permutable if $$ \rho $$ ◦ σ = σ ◦ $$ \rho $$ for any pair of congruences $$ \rho $$ , σ on S. A local automorphism of the semigroup S is defined as an isomorphism between two subsemigroups of this semigroup. The set of all local automorphisms of a semigroup S with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a classification of all finite nilsemigroups for which the inverse monoid of local automorphisms is permutable.

There are two families of element formulation for inverse elastostatic analysis in the literature. They employ different computation procedures at the element and material levels. It has been suggested that one of them can preserve material library while the other requires redeveloping the material functions. In this article, we show that these two formulations are completely equivalent. Under certain conditions both can preserve the material library while under other conditions neither can. We show that the modification of material functions is caused by the need of updating material symmetry in the inverse solution process. Several theoretical results were obtained while establishing the equivalence, including an identity relating the inverse and forward tangent tensors for isotropic materials. A comprehensive documentation of these two formulations is provided.

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.

We present a stability analysis of Gohberg–Semencul–Trench type formulae for the Moore–Penrose and group inverses of singular Toeplitz matrices. We develop a fast algorithm for the computation of the Moore–Penrose inverse based on a Gohberg–Semencul–Trench type formula and the LSQR method. For the group inverse, the DGMRES method is used to perform the fast computation. Numerical tests show that the fast algorithms designed here are at least as good as the known Newton iteration.

In this paper we study the fundamental group of inverse limits, obtained by upper semi-continuous set valued functions. We present a number of crucial examples which demonstrate the technical difficulties, related to the control of the fundamental group in the inverse limit. Furthermore, these examples realize some important groups as the fundamental groups of inverse limits: free groups and the Hawaiian Earring group. On the other hand, we introduce the right shift of a loop in the inverse limit and prove that the fundamental group of an inverse limit, which is a one-dimensional Peano continuum, is often trivial or uncountable.

The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.

Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore-Penrose inverse, and core inverse. Some properties of 2-EP are also given., This research was supported by the National Natural Science Foundation of China (No. 11371089), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Fundamental Research Funds for the Central Universities and the Foundation of Graduate Innovation Program of Jiangsu Province (No. KYZZ15-0049), the FEDER Funds through “Programa Operacional Factores de Competitividade-COMPETE”, the Portuguese Funds through FCT-“Fundação para a Ciência e a Tecnologia”, within the project UID-MAT-00013/2013.

Se presenta una propuesta didáctica para evidenciar las propiedades de la multiplicación de números reales mediante el uso de las homotecias como herramienta geométrica. Esta propuesta está dirigida a un grupo de 28 estudiantes de grado octavo de la Institución Educativa Departamental Domingo Savio, que se ubica en la zona urbana del Municipio de Guasca Cundinamarca, de carácter oficial, público y mixto. La enseñanza de las propiedades de la multiplicación de números reales se limita a un método tradicional en el cual se suelen dar muchos ejemplos donde dichas propiedades se cumplen, pero no se justifican, generando que a muchos estudiantes se les dificulte entender y diferenciar estas propiedades. Para el diseño e implementación de esta unidad didáctica se asume el aprendizaje significativo de Ausubel como referente didáctico y además, la presentación de las homotecias dada en [Papy, 1970] como herramienta geométrica, se estudian las propiedades de la composición de homotecias, se hace uso de la biyección entre los puntos de una recta y el conjunto de números reales (la recta real) y con esto se define la multiplicación de números reales mediante la composición de ciertas homotecias, para dar paso a la evidencia geométrica de las propiedades de la multiplicación. Por otra parte, se hace una revisión histórica, epistemológica y teórica de los conceptos, definiciones, axiomas y teoremas involucrados en el propósito del trabajo, para el diseño de las actividades a desarrollar con los estudiantes. Abstract: A didactic proposal is presented to show the properties of real numbers multiplication through the use of homotheties as a geometric tool. This proposal is designed for a group of 28 eighth-grade students of the Institución Educativa Departamental Domingo Savio, which is located in the urban area of the municipality of Guasca Cundinamarca; it is an official, public and mixed school. The teaching of the properties of real numbers multiplication is limited to a traditional method in which is usual to give examples where the properties are satisfied but they are not justified, so for many students is difficult to understand and to differentiate these properties. For the design and implementation of this didactic unit it is assumed the significant learning by Ausubel as a didactic reference and besides, it is considered the presentation of the homotheties given in [Papy, 1970] as a geometric tool. We study the properties of the composition of homotheties; the bijective application between the points of the line and the real numbers is used to define the multiplication of real numbers through the composition of certain homotheties, to obtain a geometric evidence of the properties of multiplication. On the other hand, it is presented a historical, epistemological and theoretical review of the concepts, definitions, axioms and theories involved in the purpose of the work, for the design of the activities to develop with the students. Maestría

In this paper, we consider the conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices. It is proved that the inverse of these special matrices can be expressed as the sum of products of lower and upper triangular matrices. Firstly, we get access to the explicit inverse of conjugate-Toeplitz matrix. Secondly, the decomposition of the inverse is obtained. Similarly, the formulae and the decomposition on inverse of conjugate-Hankel are provided. Thirdly, the stability of the inverse formulae of CT and CH matrices are discussed. Finally, examples are provided to verify the feasibility of the algorithms provided in this paper.

In this paper, we define the Moore–Penrose inverse of tensors with the Einstein product, and the explicit formulas of the Moore–Penrose inverse of some block tensors are obtained. The general solutions of some multilinear systems are given and we also give the minimum-norm least-square solution of some multilinear systems using the Moore–Penrose inverse of tensors.

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix., The authors are highly grateful to the referee for valuable comments which led to improvements of the paper. In particular, Remarks 3.2 and 3.4 were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for supporting him to purse his further study in University of Minho, Portugal. Pedro Patr´ıcio and Yulin Zhang were financed by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, through the Project PEst-OE/MAT/UI0013/2014. Jianlong Chen and Huihui Zhu were supported by the National Natural Science Foundation of China (No. 11201063 and No. 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Foundation of Graduate Innovation Program of Jiangsu Province(No. CXLX13-072), the Scientific Research Foundation of Graduate School of Southeast University and the Fundamental Research Funds for the Central Universities (No. 22420135011).

Let be a unital -algebra, a, x and y are elements in . In this paper, we present the expression of the Moore-Penrose inverse and the group inverse of a- under the conditions , respectively.

In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-regularity and left (right) $*$-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed., The authors are highly grateful to the referee for valuable comments which led to improvements of this paper. In particular, Corollaries 2.5, 2.6 and 3.6, Remarks 2.13 and 3.10 and the final remark (ii) were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for giving him a purse for his further study in University of Minho, Portugal. Jianlong Chen and Huihui Zhu are financed by the National Natural Science Foundation of China (No. 11201063 and No. 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Foundation of Graduate Innovation Program of Jiangsu Province(No. CXLX13-072), the Scientific Research Foundation of Graduate School of Southeast University and the Fundamental Research Funds for the Central Universities (No. 22420135011). Pedro Patr´ıcio is financed by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, through the Project PEst-OE/MAT/UI0013/2014.

The half-factorial, locally right Garside monoid of cofinite monotone sequences on $${\mathbb {N}}^{*}$$ is considered in this paper. This is a right cancellative, left rigid Mobius monoid, and this monoid directs us to the inverse monoid of cofinite monotone partial bijections on $${\mathbb {N}}^{*}$$ . In a certain sense, the polycyclic inverse monoid is a 0-analogue of this derived inverse monoid. Similarities and differences between these two inverse monoids are presented involving Mobius categories, Leech’s construction, breaking processes, gauge inverse submonoids, etc.

The 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, Dolžan 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.

We follow the idea to find the solution of quadratic minimization problems restricted by linear constraints and to investigate underlying generalized inverses. A generalization of this approach leads to a new kind of outer generalized inverse, called the -minimal -constrained inverse of (called the minimal inverse of shortly), with respect to a positive semidefinite/definite matrix and a singular matrix . The introduced generalized inverse is related with the minimization of the quadratic form under the constraint set defined by the consistent equation . In this way, the generalized inverse arising from this constrained quadratic minimization problem gives a minimal T-semi-norm/norm solution of the system of linear equations . The range and the null space of defined outer inverse are determined. Computational algorithms are defined and numerical experiments related with properties of the minimal inverse are also presented.

Let $M= \left[ \begin{array}{cc} A& B \\ C& O \end{array} \right]$ be a complex square matrix where A is square. When BCB^{\Omega} =0, rank(BC) = rank(B) and the group inverse of $\left[ \begin{array}{cc} B^{\Omega} A B^{\Omega} & 0 \\ CB^{\Omega} & 0 \right]$ exists, the group inverse of M exists if and only if rank(BC + A)B^{\Omega}AB^{\Omega})^{\pi}B^{\Omega}A)= rank(B). In this case, a representation of $M^#$ in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,−)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S^2GI, if the group inverse of each matrix \bar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given.

For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.

We give a monoid presentation in terms of generators and define relations for the partial analogue of the finite dual symmetric inverse monoid.

The internal Zappa-Szep products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szep products of semigroups. We consider the structure of the internal Zappa-Szep product as an enlargement. We show how rectangular band can be described as the Zappa-Szep product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szep product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szep product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szep product of a semilattice E and a group G by constructing an inductive groupoid.

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid., Comment: v2: Streamlined presentation and added results for the planar rook monoid, partial cyclic shift monoid, and partial rotation monoid. 24 pages

Let R be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ⁎-ring case. It is shown that the group, Moore–Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in R are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of ( b , c ) -inverses.

An inverse semigroup S is pure if e = e 2, a ∈ S, e < a implies a 2 = a; it is cryptic if Green’s relation H on S is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-V varieties, for varieties V of inverse semigroups considered.

The method for finding inverse element by the module is based on the stepwise addition of residue is developed in this paper. This method allows avoiding performing the complex arithmetic operations and implementing calculations on numbers much lower bit compared to the classical approach based on the Euclid's algorithm and its consequences. Analytical expressions of time complexity characteristics are obtained and their graphic depending is built. Software and hardware modules of specified methods on the basis of environment of the development AldecActive-HDL 9.1 are implemented and studies of their temporal characteristics that point to the advantages of the proposed method are conducted.

In this paper, we introduce a new concept called left (right) g-MP inverse in a *-monoid. The relations of this type of generalized inverse with left inverse along an element are investigated. Also, the reverse order law for the inverse along an element is studied. Then, the existence criteria and formulae of the inverse of the product of triple elements along an element are investigated in a monoid. Finally, we further study left and right g-MP inverses, the inverse along an element in the context of rings., This research is supported by the National Natural Science Foundation of China [No. 11371089]; the Specialized Research Fund for the Doctoral Program of Higher Education [No. 20120092110020]; the Natural Science Foundation of Jiangsu Province [No. BK20141327]; the Scientific Innovation Research of College Graduates in Jiangsu Province [No. CXLX13-072]; the Scientific Research Foundation of Graduate School of Southeast University, the FEDER Funds through 'Programa Operacional Factores de Competitividade-COMPETE', the Portuguese Funds through FCT - 'Fundacao para a Ciencia e a Tecnologia', within the project [UID/MAT/00013/2013]., info:eu-repo/semantics/publishedVersion

While Theorem 4.8.9 implies that the type monoid of a Boolean inverse semigroup S can be any countable conical refinement monoid, there are situations in which the structure of S impacts greatly the one of \(\mathop{\mathrm{Typ}}\nolimits S\). A basic illustration of this is given by the class of AF inverse semigroups , introduced in Lawson and Scott [77], which is the Boolean inverse semigroup version of the class of AF C*-algebras. Another Boolean inverse semigroup version of a class of C*-algebras, which we will not consider here, is given by the Cuntz inverse monoids studied in Lawson and Scott [77, § 3].

This chapter presents a brief introduction to the generalized inverse of matrix, which is needed in the following expositions. This introduction includes the left inverse and right inverse, the Moore-Penrose inverse, the minimization approach to solve an algebraic matrix equation, the full rank decomposition theorem, the least square solution to an algebraic matrix equation, and the singular value decomposition.

Tarski investigates in [109] partial commutative monoids constructed from partial bijections on a given set. In Kudryavtseva et al. [71], this study is conveniently formalized in the language of inverse semigroups. Further connections can be found in works on K-theory of rings, such as Ara and Exel [7].