number and length of attractors in a critical kauffman model with connectivity one the kauffman model describes a system of randomly connected nodes with dynamics based on boolean update functions though it is a simple model it exhibits very complex behavior for critical parameter values at the boundary between a frozen and a disordered phase and is therefore used for studies of real network problems we prove here that the mean number and mean length of attractors in critical random boolean networks with connectivity one both increase faster than any power law with network size we derive these results by generating the networks through a growth process and by calculating lower bounds

state analysis of boolean control networks with impulsive and uncertain disturbances in this paper we put forward a method for the state analysis of boolean control networks with impulsive and uncertain disturbances in detail we introduce the probability in the analysis of the state of boolean networks mathematically the stp as the analytical tool has be used for the modeling of uncertainty in addition we give a mathematical proof for the model lastly we conduct some simulations which validate the efficiency for this model

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

a matrix convexity approach to some celebrated quantum inequalities abstract some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the hansen pedersen jensen inequalities a general argument is given using matrix perspectives of operator convex functions a matrix analogue of marechal s extended perspectives provides additional inequalities including a p q 1 result of lieb

synchronization of boolean networks with time delays abstract this paper studies analytically synchronization of two deterministic boolean networks with time delays which are unidirectionally coupled in the drive response configuration necessary and sufficient conditions of synchronization are established based on the algebraic representations of logical dynamics in terms of the semi tensor product of matrices the synchronization conditions show that for drive response boolean networks different relations between the inherent state delay and the unidirectional coupling delay may result in distinct synchronization phenomena and both kinds of time delays that ensure synchronization if existent are not unique examples are worked out to illustrate the present results

matrix inequalities in statistical mechanics some matrix inequalities used in statistical mechanics are presented a straightforward proof of the thermodynamic inequality is given and its equivalence to the peierls bogoliubov inequality is shown

operator entropy and fidelity associated with the geometric mean abstract we consider operator versions of quantum tsallis relative entropy that involve the weighted matrix geometric mean and establish relationships with the fidelity of a pure state and an arbitrary state we define the symmetrized operator entropy based on the tsallis relative operator entropy and study its properties

on sub and superadditive vector valued maps with applications to group majorization in this paper motivated by results in s s dragomir quasilinearity of some functionals associated with monotonic convex functions rgmia res rep coll 14 2011 artcle 24 13 pp and m niezgoda cone orderings group majorizations and similarly separable sequences linear algebra appl 436 3 2012 pp 579 594 we investigate maps involving quotient p p with an 1 subadditive operator and a positive superadditive functional we pay special attention to vector valued operators we study the subadditivity of the map p p f p p with 1 increasing convex f next we give applications for group majorization preorders we establish some preorder preservation results for the map p p p with a group majorization g and a combination 1 2 of preorders 1 and 2 in consequence we show that the function p f p p is g increasing

shannon like inequalities for f connections of positive linear maps and positive operators abstract in this paper we investigate a map t b t f t b called an f connection induced by an operator convex concave function f where t denotes a reflexive generalized inverse of a positive linear map t between unital c algebras a and b of hilbert space operators and b b some special cases of f connection with invertible t are related to geometric operator mean relative operator entropy and csiszar operator f divergence we formulate conditions under which the inequality f 1 i k 1 n t k f t k b k holds where t k a b are positive linear maps and b k ran t k in particular the following shannon like inequality 0 k 1 n t k f t k b k is shown for an operator convex function f with f 1 0 the obtained results are specified for csiszar operator f divergence some recent results by isa et al 2013 15 are recovered

the ando hiai inequalities for the solution of the generalized karcher equation and related results abstract in this paper we shall give a concrete relation between the generalized karcher equation and operator means as its solution next we shall show two types of the ando hiai inequalities for the solution of the generalized karcher equation in this discussion we also give a property of relative operator entropy

satisfiability in many valued sentential logic is np complete abstract cook s np completeness theorem is extended to all many valued sentential logics including the infinite valued calculus of ukasiewicz

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

state feedback stabilization for boolean control networks state feedback stabilization for boolean control networks is investigated in this technical note based on the algebraic representation of logical dynamics in terms of the semi tensor product of matrices a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a general control design approach is proposed when global stabilization is feasible via state feedback instead of designing the logical form of a stabilizing feedback law directly we first construct its algebraic representation and then convert the algebraic representation back to the logical form an example is worked out to illustrate the proposed design procedure

two reverse inequalities associated with tsallis relative operator entropy via generalized kantorovich constant and their applications recently tsallis relative operator entropy tp a b and tsallis relative entropy dp a b are discussed by furuichi yanagi kuriyama we shall show two reverse inequalities involving tsallis relative operator entropy tp a b via generalized kantorovich constant k p as some applications of two reverse inequalities we shall show two trace reverse inequalities involving tr tp a b and dp a b and also a known reverse trace inequality involving the relative operator entropy s a b by fujii kamei and the umegaki relative entropy s a b is shown as a simple corollary we show the following result let a and b be strictly positive operators on a hilbert space h such that m1 i a m1 i 0 and m2 i b m2 i 0 put m m2m1 m m2m1 h mm m1m2m1m2 1 and p 0 1 let be normalized positive linear map on b h then the following inequalities hold i 1 k p p a p b tp a b tp a b tp a b and ii f p a tp a b tp a b tp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and k p 0 1 and f p mpphp hh 11 k p 1p 1 0 in addition let a and b be strictly positive definite matrices iii 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b and iv f p tr a dp a b tr tp a b dp a b in particular both iii and iv yield the following known result logs 1 tr a s a b tr s a b s a b where s 1 h1h 1elogh1h 1 is said to be the specht ratio and s 1 1

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

the morita equivalence between mv algebras and lattice ordered abelian groups with strong unit abstract we show that the theory of mv algebras is morita equivalent to that of abelian l groups with strong unit this generalizes the well known equivalence between the categories of set based models of the two theories established by d mundici in 1986 and allows to transfer properties and results across them by using the methods of topos theory we discuss several applications including a sheaf theoretic version of mundici s equivalence and a bijective correspondence between the geometric theory extensions of the two theories

finitely presented lattice ordered abelian groups with order unit abstract let g be an l group which is short for lattice ordered abelian group baker and beynon proved that g is finitely presented iff it is finitely generated and projective in the category u of unital l groups those l groups having a distinguished order unit u only the direction holds in general we show that a unital l group g u is finitely presented iff it has a basis a large class of projectives is constructed from bases having special properties

word problems in elliott monoids abstract algorithmic issues concerning elliott local semigroups are seldom considered in the literature although these combinatorial structures completely classify af algebras in general the addition operation of an elliott local semigroup is partial but for every af algebra b whose murray von neumann order of projections is a lattice this operation is uniquely extendible to the addition of an involutive monoid e b let m 1 be the farey af algebra introduced by the present author in 1988 and rediscovered by f boca in 2008 the freeness properties of the involutive monoid e m 1 yield a natural word problem for every af algebra b with singly generated e b because b is automatically a quotient of m 1 given two formulas and in the language of involutive monoids the problem asks to decide whether and code the same equivalence of projections of b this mimics the classical definition of the word problem of a group presented by generators and relations we show that the word problem of m 1 is solvable in polynomial time and so is the word problem of the behncke leptin algebras a n k and of the effros shen algebras f for 0 1 q a real algebraic number or 1 e we construct a quotient of m 1 having a godel incomplete word problem and show that no primitive quotient of m 1 is godel incomplete

simple bratteli diagrams with a godel incomplete c equivalence problem an abstract simplicial complex is a finite family of subsets of a finite set closed under subsets every abstract simplicial complex c naturally determines a bratteli diagram and a stable af algebra a c consider the following problem input a pair of abstract simplicial complexes c and c question is a c isomorphic to a c we show that this problem is godel incomplete i e it is recursively enumerable but not decidable this result is in sharp contrast with the recent decidability result by bratteli jorgensen kim and roush for the isomorphism problem of stable af algebras arising from the iteration of the same positive integer matrix for the proof we use a combinatorial variant of the de concini procesi theorem for toric varieties together with the baker beynon duality theory for lattice ordered abelian groups markov s undecidability result and elliott s classification theory for af algebras

farey stellar subdivisions ultrasimplicial groups and k0 of af c algebras abstract the theory will i believe admit of being extended to solid reticulations formed by the intersections of three systems of equidistant parallel planes determinants of the third order between the three coordinates of successive points replacing the pq p q of the plane theory

a minimum time control for boolean control networks with impulsive disturbances this paper investigates a mayer type optimal control and minimum time control of a boolean control network bcn with impulsive disturbances using the semi tensor product the bcn with impulsive disturbances is converted into algebraic discrete time impulsive dynamic systems and several necessary conditions for optimality are derived then we consider the problem of steering a bcn with impulsive disturbances from a given initial state to a desired state in minimal time and a necessary condition stated in the form of maximum principle is obtained for a control to be time optimal it shows that the impulsive disturbances play an important role in the optimal control problem for bcns at last a biological example is given to illustrate the effectiveness and advantage of the obtained results

parametric extensions of shannon inequality and its reverse one in hilbert space operators we shall state the following parametric extensions of shannon inequality and its reverse one in hilbert space operators let p 0 1 and also let a1 a2 an and b1 b2 bn be two sequences of strictly positive operators on a hilbert space h such that j 1naj pbj i then j 1nsp 1 aj bj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp aj bj log j 1n aj p 1bj i j 1naj pbj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp 1 aj bj where sq a b a12a 12ba 12qloga 12ba 12a12 for a 0 b 0 and any real number q and a qb a12a 12ba 12qa12 for a 0 b 0 and any real number q in particular if j 1naj j 1nbj i then j 1ns2 aj bj j 1nbjaj 1bjlog j 1nbjaj 1bj log j 1nbjaj 1bj j 1ns1 aj bj 0 j 1ns aj bj log j 1najbj 1aj j 1najbj 1ajlog j 1najbj 1aj j 1ns 1 aj bj where s a b s0 a b a12loga 12ba 12a12 which is the relative operator entropy of a 0 and b 0 our results can be considered as parametric extensions of the following celebrated shannon inequality ann math statistics 22 1951 79 bull syst tech j 27 1948 379 pitman monographs and surveys in pure and applied mathematics 97 addison wesley longman 1998 p 233 which is very useful and so famous in information theory let a1 a2 an and b1 b2 bn be two probability vectors then 0 j 1najlogbj j 1najlogaj see inequalities 2 4 of corollary 2 4

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

on robust control invariance of boolean control networks this paper investigates the robust control invariance of boolean control networks bcns via the semi tensor product of matrices firstly based on an algebraic state space representation of bcns two necessary and sufficient conditions are presented to check whether or not a given set is a robust control invariant set under a given state feedback controller secondly by defining a series of suitable sets all possible state feedback gain matrices under which a given set is a robust control invariant set are characterized an illustrative example is presented to demonstrate the obtained new results

on operator inequalities of some relative operator entropies abstract in our recent paper we introduced the notions of relative operator entropy and tsallis relative operator entropy as a parameter extensions of relative operator entropy and tsallis relative operator entropy in this paper we give upper and lower bounds of these new notions according to operator geometric mean introduced in nikoufar et al 2013 14

on jensen renyi and jeffreys renyi type f divergences induced by convex functions abstract in this paper renyi type f divergences like jensen renyi and jeffreys renyi are studied for convex functions some comparison theorems for two divergences are provided the obtained results imply for introduced divergences some new inequalities corresponding to nonnegative convex functions some examples of known entropies are introduced in order to show a few applications of new divergences

stability and stabilization of boolean networks with impulsive effects abstract this paper investigates the stability and stabilization of boolean networks with impulsive effects after giving a survey on semi tensor product of matrices we convert a boolean network with impulsive effects into impulsive discrete time dynamics then some necessary and sufficient conditions are given for the stability and stabilization of boolean networks with impulsive effects finally examples are provided to illustrate the efficiency of the obtained results

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

generalized matrix version of reverse holder inequality abstract a generalized matrix version of reverse cauchy schwarz holder inequality is proved this includes the recent results proved by bourin fujii lee niezgoda and seo

sheaf representations of mv algebras and lattice ordered abelian groups via duality abstract we study representations of mv algebras equivalently unital lattice ordered abelian groups through the lens of stone priestley duality using canonical extensions as an essential tool specifically the theory of canonical extensions implies that the stone priestley dual spaces of mv algebras carry the structure of topological partial commutative ordered semigroups we use this structure to obtain two different decompositions of such spaces one indexed over the prime mv spectrum the other over the maximal mv spectrum these decompositions yield sheaf representations of mv algebras using a new and purely duality theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces importantly the proofs of the mv algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features 1 we use only basic algebraic facts about mv algebras 2 we show that the two aforementioned sheaf representations are special cases of a common result with potential for generalizations and 3 we show that these results are strongly related to the structure of the stone priestley duals of mv algebras in addition using our analysis of these decompositions we prove that mv algebras with isomorphic underlying lattices have homeomorphic maximal mv spectra this result is an mv algebraic generalization of a classical theorem by kaplansky stating that two compact hausdorff spaces are homeomorphic if and only if the lattices of continuous 0 1 valued functions on the spaces are isomorphic

generalized shannon inequalities based on tsallis relative operator entropy abstract tsallis relative operator entropy is defined and then its properties are given shannon inequality and its reverse one in hilbert space operators derived by furuta linear algebra apll 381 2004 219 are extended in terms of the parameter of the tsallis relative operator entropy moreover the generalized tsallis relative operator entropy is introduced and then several operator inequalities are derived

reverse inequalities involving two relative operator entropies and two relative entropies we shall discuss relation among tsallis relative operator entropy tp a b the relative operator entropy s a b by j i fujii kamei the tsallis relative entropy dp a b by furuichi yanagi kuriyama and the umegaki relative entropy s a b we show the following result let a and b be strictly positive definite matrices such that m1i a m1i 0 and m2i b m2i 0 put h m1m2m1m2 1 and p 0 1 then the following inequalities hold 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and the first inequality is the reverse one of the second known inequalty in particular logs 1 tr a s a b tr s a b s a b where s 1 is the specht ratio defined by s 1 h1h 1elogh1h 1 and the first inequality is the reverse one of the second known inequalty it is known that k p 0 1 for p 0 1 and s 1 1

every abelian l group is ultrasimplicial abstract a partially ordered abelian group g is said to be ultrasimplicial if for every finite set p of positive elements of g there is a finite set b of positive elements which are linearly independent in the z module g and such that p belongs to the monoid generated by b in this paper we prove the result stated in the title

generalizations of crooks and lin s results on jeffreys csiszar and jensen csiszar f divergences abstract in this paper jeffreys csiszar and jensen csiszar f divergences are introduced and studied some bounds of crooks and lin types for such divergences are provided to this end the concavity of the composition of monotone functions is discussed refinements of the original inequalities by crooks and popescu et al are given

global stability and stabilization of switched boolean network with impulsive effects the present paper investigates the global stability and stabilization of switched boolean network with impulsive effects by using the semi tensor product the algebraic form of the switched boolean network with impulsive effects is obtained then sufficient and necessary condition is presented for the global stability of the switched boolean network with impulsive effects furthermore sufficient and necessary conditions are provided for the stabilization of the switched boolean network with impulsive effects under two types of controls finally examples are presented to illustrate the main results

the logical content of triangular bases of fuzzy sets in łukasiewicz infinite valued logic abstract continuing to pursue a research direction that we already explored in connection with godel dummett logic and ruspini partitions we show here that ukasiewicz logic is able to express the notion of pseudo triangular basis of fuzzy sets a mild weakening of the standard notion of triangular basis en route to our main result we obtain an elementary logic independent characterisation of triangular bases of fuzzy sets

a linear representation of dynamics of boolean networks a new matrix product called semi tensor product of matrices is reviewed using it a matrix expression of logic is proposed where a logical variable is expressed as a vector a logical function is expressed as a multiple linear mapping under this framework a boolean network equation is converted into an equivalent algebraic form as a conventional discrete time linear system analyzing the transition matrix of the linear system formulas are obtained to show a the number of fixed points b the numbers of cycles of different lengths c transient period for all points to enter the set of attractors and d basin of each attractor the corresponding algorithms are developed and used to some examples

generalizations of operator shannon inequality based on tsallis and renyi relative entropies abstract recently we obtained relations among relative operator entropies of sequences for example for relative operator entropy s a b renyi relative operator entropy i t a b and tsallis relative operator entropy t t a b of sequences of strictly positive operators a a 1 a n and b b 1 b n such that i 1 n a i i 1 n b i i s a b i t a b t t a b 0 holds for 0 t 1 this is an extension of operator version of shannon inequality briefly operator shannon inequality discussed by furuta and yanagi kuriyama furuichi in this paper we shall obtain two generalizations of this inequality by considering generalizations of relative operator entropies of sequences

lattice ordered abelian groups and schauder bases of unimodular fans baker beynon duality theory yields a concrete representation of any finitely generated projective abelian lattice ordered group g in terms of piecewise linear homogeneous functions with integer coefficients defined over the support e of a fan a unimodular fan a over s determines a schauder basis of g its elements are the minimal positive free generators of the pointwise ordered group of a linear support functions conversely a schauder basis h of g determines a unimodular fan over its maximal cones are the domains of linearity of the elements of h the main purpose of this paper is to give various representation free characterisations of schauder bases the latter jointly with the de concini procesi starring technique will be used to give novel characterisations of finitely generated projective abelian lattice ordered groups for instance g is finitely generated projective iff it can be presented by a purely lattice theoretical word

stabilization controllability and optimal control of boolean networks with impulsive effects and state constraints the present note investigates the stabilization controllability and optimal control problem of boolean networks with impulsive effects and state constraints by using the semi tensor product method the algebraic form of the boolean networks with impulsive effects and state constraints is derived the stabilization and controllability issues of the systems are investigated and some necessary and sufficient conditions are obtained in addition the mayer type optimal control problem is also studied and algorithms are provided to design the control sequence furthermore examples are given to illustrate the main results

classes of ultrasimplicial lattice ordered abelian groups abstract a lattice ordered abelian group is called ultrasimplicial iff every finite set of positive elements belongs to the monoid generated by some finite set of positive z independent elements this property originates from elliott s classification of af c algebras using fans and their desingularizations it is proved that the ultrasimplicial property holds for every n generated archimedean l group whose maximal l ideals of rank n are dense as a corollary we obtain simpler proofs of results respectively by elliott and by the present author stating that totally ordered abelian groups as well as free l groups are ultrasimplicial

path of quasi means as a geodesic abstract as a generalization of the hiai petz geometries we discuss two types of them where the geodesics are the quasi arithmetic means and the quasi geometric means respectively each derivative of such a geodesic might determine a new relative operator entropy also in these cases the finsler metric can be induced by each unitarily invariant norm if the norm is strictly convex then the geodesic is the shortest we also give examples of the shortest paths which are not the geodesics when the finsler metrics are induced by the ky fan k norms

disturbance decoupling of boolean control networks disturbance decoupling problem ddp of boolean control networks is considered using semi tensor product of matrices and the matrix expression of logical functions a working procedure is proposed to solve the problem this procedure consists of two key design steps first how to convert a system into an output friendly coordinate frame an algorithm is provided to calculate the output friendly subspaces secondly it was shown how to find proper controllers to solve the problem if it is solvable a state variable separation form is introduced to guide the design of controllers based on the design technique necessary and sufficient conditions are obtained for the solvability of ddp

matrix trace inequalities on tsallis relative entropy of negative order abstract in this paper we show matrix trace inequalities on the quantum tsallis relative entropy of negative order which includes the quasi geometric mean of positive definite matrices moreover we show an estimate of the difference between two tsallis relative entropies of negative order

a note on operator inequalities of tsallis relative operator entropy tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized shannon inequalities were shown in the previous paper after the review of some fundamental properties of tsallis relative operator entropy some operator inequalities related to tsallis relative operator entropy are shown in the present paper our inequalities give the upper and lower bounds of tsallis relative operator entropy the operator equality on tsallis relative operator entropy is also shown by considering the tensor product this relation generalizes the pseudoadditivity for tsallis entropy as a corollary of our operator equality derived from the tensor product manipulation we show several operator inequalities including the superadditivity and the subadditivity for tsallis relative operator entropy our results are generalizations of the superadditivity and the subadditivity for tsallis entropy