Von Neumann Regular Cellular Automata

For any group $G$ and any set $A$, a cellular automaton (CA) is a transformation of the configuration space $A^G$ defined via a finite memory set and a local function. Let $\text{CA}(G;A)$ be the monoid of all CA over $A^G$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element $\tau \in \text{CA}(G;A)$ is von Neumann regular (or simply regular) if there exists $\sigma \in \text{CA}(G;A)$ such that $\tau \circ \sigma \circ \tau = \tau$ and $\sigma \circ \tau \circ \sigma = \sigma$, where $\circ$ is the composition of functions. Such an element $\sigma$ is called a generalised inverse of $\tau$. The monoid $\text{CA}(G;A)$ itself is regular if all its elements are regular. We establish that $\text{CA}(G;A)$ is regular if and only if $\vert G \vert = 1$ or $\vert A \vert = 1$, and we characterise all regular elements in $\text{CA}(G;A)$ when $G$ and $A$ are both finite. Furthermore, we study regular linear CA when $A= V$ is a vector space over a field $\mathbb{F}$; in particular, we show that every regular linear CA is invertible when $G$ is torsion-free elementary amenable (e.g. when $G=\mathbb{Z}^d, \ d \in \mathbb{N}$) and $V=\mathbb{F}$, and that every linear CA is regular when $V$ is finite-dimensional and $G$ is locally finite with $\text{Char}(\mathbb{F}) \nmid o(g)$ for all $g \in G$.Comment: 10 pages. Theorem 5 corrected from previous versions, in A. Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer, 201

Centralizer's applications to the (b, c)-inverses in rings

[EN] We give several conditions in order that the absorption law for one sided (b,c)-inverses in rings holds. Also, by using centralizers, we obtain the absorption law for the (b,c)-inverse and the reverse order law of the (b,c)-inverse in rings. As applications, we obtain the related results for the inverse along an element, Moore-Penrose inverse, Drazin inverse, group inverse and core inverse.This research is supported by the National Natural Science Foundation of China (no. 11771076 and no. 11871301). The first author is grateful to China Scholarship Council for giving him a scholarship for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; Benítez López, J.; Wang, D. (2019). Centralizer's applications to the (b, c)-inverses in rings. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. 113(3):1739-1746. https://doi.org/10.1007/s13398-018-0574-0S173917461133Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Benítez, J., Boasso, E.: The inverse along an element in rings with an involution, Banach algebras and $$C^*$$ C ∗ -algebras. Linear Multilinear Algebra 65(2), 284–299 (2017)Benítez, J., Boasso, E., Jin, H.W.: On one-sided $$(B, C)$$ ( B , C ) -inverses of arbitrary matrices. Electron. J. Linear Algebra 32, 391–422 (2017)Boasso, E., Kantún-Montiel, G.: The $$(b, c)$$ ( b , c ) -inverses in rings and in the Banach context. Mediterr. J. Math. 14, 112 (2017)Chen, Q.G., Wang, D.G.: A class of coquasitriangular Hopf group algebras. Comm. Algebra 44(1), 310–335 (2016)Chen, J.L., Ke, Y.Y., Mosić, D.: The reverse order law of the $$(b, c)$$ ( b , c ) -inverse in semigroups. Acta Math. Hung. 151(1), 181–198 (2017)Deng, C.Y.: Reverse order law for the group inverses. J. Math. Anal. Appl. 382(2), 663–671 (2011)Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)Drazin, M.P.: Left and right generalized inverses. Linear Algebra Appl. 510, 64–78 (2016)Jin, H.W., Benítez, J.: The absorption laws for the generalized inverses in rings. Electron. J. Linear Algebra 30, 827–842 (2015)Johnson, B.E.: An introduction to the theory of centralizers. Proc. Lond. Math. Soc. 14, 299–320 (1964)Ke, Y.Y., Cvetković-Ilić, D.S., Chen, J.L., Višnjić, J.: New results on $$(b, c)$$ ( b , c ) -inverses. Linear Multilinear Algebra 66(3), 447–458 (2018)Ke Y.Y., Višnjić J., Chen J.L.: One sided $$(b,c)$$ ( b , c ) -inverse in rings (2016). arXiv:1607.06230v1Liu, X.J., Jin, H.W., Cvetković-Ilić, D.S.: The absorption laws for the generalized inverses. Appl. Math. Comput. 219, 2053–2059 (2012)Mary, X.: On generalized inverse and Green’s relations. Linear Algebra Appl. 434, 1836–1844 (2011)Mary, X., Patrício, P.: Generalized inverses modulo $$\cal{H}$$ H in semigroups and rings. Linear Multilinear Algebra 61(8), 1130–1135 (2013)Mosić, D., Cvetković-Ilić, D.S.: Reverse order law for the Moore-Penrose inverse in $$C^*$$ C ∗ -algebras. Electron. J. Linear Algebra 22, 92–111 (2011)Rakić, D.S.: A note on Rao and Mitra’s constrained inverse and Drazin’s $$(b, c)$$ ( b , c ) -inverse. Linear Algebra Appl. 523, 102–108 (2017)Rakić, D.S., Dinčić, N.Č., Djordjević, D.S.: Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)Wang, L., Castro-González, N., Chen, J.L.: Characterizations of outer generalized inverses. Can. Math. Bull. 60(4), 861–871 (2017)Wei, Y.M.: A characterization and representation of the generalized inverse $$A^{(2)}_{T, S}$$ A T , S ( 2 ) and its applications. Linear Algebra Appl. 280, 87–96 (1998)Xu, S.Z., Benítez, J.: Existence criteria and expressions of the $$(b, c)$$ ( b , c ) -inverse in rings and its applications. Mediterr. J. Math. 15, 14 (2018)Zhu, H.H., Chen, J.L., Patrício, P.: Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64(3), 393–403 (2016)Zhu, H.H., Chen, J.L., Patrício, P.: Reverse order law for the inverse along an element. Linear Multilinear Algebra 65, 166–177 (2017)Zhu, H.H., Chen, J.L., Patrício, P., Mary, X.: Centralizer’s applications to the inverse along an element. Appl. Math. Comput. 315, 27–33 (2017)Zhu, H.H., Zhang, X.X., Chen, J.L.: Centralizers and their applications to generalized inverses. Linear Algebra Appl. 458, 291–300 (2014

The one-sided inverse along an element in semigroups and rings

The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that a∈ S † ⋂ S # if and only if (a∗a)k is invertible along a if and only if (aa∗)k is invertible along a in a ∗ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along aa ∗ coincides with the core inverse of a under the condition a∈ S { 1 , 4 } in a ∗ -monoid S.FCT - Fuel Cell Technologies Program(CXLX13-072)This research was 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) and the Foundation of Graduate Innovation Program of Jiangsu Province (No. KYZZ15-0049).info:eu-repo/semantics/publishedVersio

A weak group inverse for rectangular matrices

[EN] In this paper, we extend the notion of weak group inverse to rectangular matrices (called WweightedWGinverse) by using the weighted core EP inverse recently investigated. This new generalized inverse also generalizes the well-known weighted group inverse given by Cline and Greville. In addition, we give several representations of the W-weighted WG inverse, and derive some characterizations and properties.First author was partially supported by UNRC (Grant PPI 18/C472) and CONICET (Grant PIP 112-201501-00433CO). Third author was partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grants DGI MTM2013-43678-P and Red de Excelencia MTM2017-90682-REDT).Ferreyra, DE.; Orquera, V.; Thome, N. (2019). A weak group inverse for rectangular matrices. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3727-3740. https://doi.org/10.1007/s13398-019-00674-9S372737401134Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. Comput. 236, 450–457 (2014)Bajodah, A.H.: Servo-constraint generalized inverse dynamics for robot manipulator control design. Int. J. Robot. Autom. 25, (2010). https://doi.org/10.2316/Journal.206.2016.1.206-3291Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear transformations. SIAM, Philadelphia (2009)Cline, R.E., Greville, T.N.E.: A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 53–62 (1980)Dajić, A., Koliha, J.J.: The weighted g-Drazin inverse for operators. J. Aust. Math. Soc. 2, 163–181 (2007)Doty, K.L., Melchiorri, C., Bonivento, C.: A theory of generalized inverses applied to robotics. Int. J. Rob. Res. 12, 1–19 (1993)Drazin, M.P.: Pseudo-inverses in associate rings and semirings. Am. Math. Mon. 65, 506–514 (1958)Ferreyra, D.E., Levis, F.E., Thome, N.: Revisiting of the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41, 265–281 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Maximal classes of matrices determining generalized inverses. Appl. Math. Comput. 333, 42–52 (2018)Gigola, S., Lebtahi, L., Thome, N.: The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem. J. Comput. Appl. Math. 291, 449–457 (2016)Hartwig, R.E.: The weighted $$*$$ ∗ -core-nilpotent decomposition. Linear Algebra Appl. 211, 101–111 (1994)Kirkland, S.J., Neumann, M.: Group inverses of M-matrices and their applications. Chapman and Hall/CRC, London (2013)Malik, S., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226, 575–580 (2014)Male $${{\check{\rm s}}}$$ s ˇ ević, B., Obradović, R., Banjac, B., Jovović, I., Makragić, M.: Application of polynomial texture mapping in process of digitalization of cultural heritage. arXiv:1312.6935 (2013). Accessed 14 June 2018Manjunatha Prasad, K., Mohana, K.S.: Core EP inverse. Linear Multilinear Algebra 62, 792–802 (2014)Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66, 1046–1053 (2018)Meng, L.S.: The DMP inverse for rectangular matrices. Filomat 31, 6015–6019 (2017)Mosić, D.: The CMP inverse for rectangular matrices. Aequaetiones Math. 92, 649–659 (2018)Penrose, R.: A generalized inverse for matrices. Proc. Cambrid. Philos. Soc. 51, 406–413 (1955)Soleimani, F., Stanimirović, P.S., Soleymani, F.: Some matrix iterations for computing generalized inverses and balancing chemical equations. Algorithms 8, 982–998 (2015)Xiao, G.Z., Shen, B.Z., Wu, C.K., Wong, C.S.: Some spectral techniques in coding theory. Discrete Math. 87, 181–186 (1991)Wang, H.: Core-EP decomposition and its applications. Linear Algebra Appl. 508, 289–300 (2016)Wang, H., Chen, J.: Weak group inverse. Open Math. 16, 1218–1232 (2018)Wei, Y.: A characterization for the $$W$$ W -weighted Drazin inverse and a Crammer rule for the $$W$$ W -weighted Drazin inverse solution. Appl. Math. Comput. 125, 303–310 (2002

Several types of one-sided partial orders in rings

In this paper, we investigate one-sided sharp partial orders and one-sided core and dual core partial orders in rings. Moreover, their characterizations and relations with other partial orders are given.This research is supported by the National Natural Science Foundation of China (No. 11801124), the Natural Science Foundation of Anhui Province (No. 1808085QA16), the Fundamental Research Funds for the Central Universities (No. JZ2018HGTB0233) and the Portuguese Funds through FCT- ‘Fundação para a Ciência e a Tecnologia’, within the project UID-MAT-00013/2013