MULTI–ACTIONS OF VECTOR SPACES ON PHASE
HYPERGROUPS OF LINEAR DIFFERENTIAL
OPERATORS
JAN CHVALINA, ŠTĚPÁN KŘEHLÍK, AND MICHAL NOVÁK
∗
Abstract. This paper is motivated by models of specific time processes. Inspired by some frequentely used modelling functions we construct hyperstructures of vectors of real functions and of linear differential operators. Then we use these as state-sets and input-sets of quasimultiautomata, i.e. of automata without output generalized into the
hyperstructure context. Finally, we construct homogeneous and heterogeneous products of such quasi-multiautomata.
Keywords: Generalization of automata without output, heterogeneous
product of automata, homogeneous product of automata, modelling
functions, linear differential operators.
MSC(2010): Primary: 20N20; Secondary: 20M35, 68Q70.
1. Introduction
The algebraic theory of automata is a widely studied classical discipline.
Apart from many studies devoted to finite automata and sequential machines in connection with theory of formal languages, infinite automata and
their generalizations have also been dealt with by many authors including
Gécseg, Peák, Bavel, Dörfler [2, 15, 16]. The infinite automata without output, also called quasi–automata, are one of the basic theoretical resources
for modelling of discrete computing systems. In fact, they are discrete modifications or“algebraic skelets” of dynamical systems. Modifications of these
concepts are described and studied in [16].
Recent rapid development of the algebraic hyperstructures theory naturally leads to the investigation of actions of hyperstructures on sets of
various objects, in particular on sets of continuous or differentiable functions. In this respect, the concept of a quasi-automaton transforms into
the concept of a quasi-multiautomaton. First steps in this area were done
by Ashrafi and Madanshekaf [1] and Massouros and Mittas [22, 23]. Later,
Chvalina proposed the construction via the GMAC condition (5.5) and together with Chvalinová, Hošková and others followed this line of research
in e.g. [7, 8, 9, 12]. Together with Deghan Nezhad, Chvalina and Hošková–
Mayerová [10] studied the issue from a broader perspective. For a deeper
inside in the topic see [7], especially its Concluding remark. Some properties
∗
Corresponding author.
1
2
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
of the GMAC-based automata have recently been studied by Račková [26].
Borzooei, Varasteh and Hasankhani [5] have recently transferred the idea of
quasi–multiautomata to the context of fuzzy (hyper)structures. The ideas
of [1] were followed e.g. by Zhan et al. [27].
The structures and hyperstructures studied in our paper are motivated
by linear differential operators which form left–hand sides of linear homogeneous differential equations, solutions of which are functions used for modelling of specific processes in electrical engineering and in kernel physics. In
our paper we investigate two types of products of quasi-multiautomata: homogeneous and heterogeneous ones. In these we adjust the ideas of Dörfler [15],
which can be traced back to Birkhoff and Lipson [4].
2. Motivation
Constructions used in our paper are motivated by differential equations
induced by specific modelling functions. Since all of these are functions of
time, we always regard t ∈ h0, ∞). Our first example is the function of the
muffled oscillations
(2.1)
y(t) = a exp(−λt) sin(bt).
Its first and second derivatives are y 0 (t) = a exp(−λt) (b
cos(bt) − λ sin(bt))
00
2
2
and y (t) = a exp(−λt) (λ − b ) sin(bt) − 2λt cos(bt) , respectively. The
relation between y 00 (t), y 0 (t) and y(t) is described by the differential equation
(2.2)
y 00 (t) + 2λy 0 (t) + (λ2 + b2 )y(t) = 0.
As another example, consider the modelling time function
(2.3)
y(t) = exp(αt) − exp(βt), α < β,
which is in fact a two parameter time signal known as multiexponential function of nuclei decay. Its derivatives are y 0 (t) = −α exp(−αt) + β exp(−βt)
and y 00 (t) = α2 exp(−αt) + β 2 exp(−βt), which leads to the following differential equation
(2.4)
y 00 (t) − (α − β)y 0 (t) + αβy(t) = 0.
Finally, we mention two functions which are solutions of differential equations in the Jacobi form. First, the Gaussian–shaped pulse signal
(2.5)
v(t) = a exp(−2πt2 ),
the first and second derivatives of which are v 0 (t) = −4aπt exp(−2πt2 ) and
v 00 (t) = 16aπ 2 t2 v(t), respectively. This leads to the differential equations in
the Jacobi form
(2.6)
v 00 (t) + (4π − 16aπ 2 t2 )v(t) = 0
with the parameter a running through a suitable number set. Second, the
Chapman-Richard’s function (CHRF)
(2.7)
y = A[1 − exp(−ct)]b ,
MULTI–ACTIONS OF VECTOR SPACES
3
one of the most common functions based on the original Bertalanffy equation derived for growth and increment of body weight. Its first and second
derivatives are y 0 = Abc[1 − exp(−ct)]b−1 exp(−ct) and y 00 = −Abc2 [1 −
exp(ct)−b
exp(−ct)]b (exp(ct)−1)
From these we obtain a second-order
2 , respectively.
linear differential equation in the Jacobi form
(2.8)
y 00 (t) +
bc2 · exp(−ct) · [b exp(−ct) − 1]
y(t) = 0.
(1 − exp(−ct))2
Further on we regard matrices and their structures motivated by the needs
of signal analysis and processing which is one of the important and fast growing areas in applications in fields such as traditional telecommunications and
AV technology, processing of measurement results and system identification
in civil or chemical engineering, biomedicine, environmental and economical
analysis. To give a few examples included e.g. in [3, 18, 19, 21]:
(1) Linear discrete systems are described by means of input/output
models. In some applications a more general state model is used.
This model works with vector input and output and describes values in chosen internal points of the system and enables its users to
transform the basic realiyation structure into structures which are
(from the point of view of the input/output correspondence) equivalent. The usual form of the state model is
(2.9)
~qn+1 = A~qn + B~xn
~yn = C~qn + D~xn ,
where ~qn = (qn1 , qn2 , . . . , qnm )T is a column vector of internal, state
variables, ~xn = (x1n , x2n , . . . , xpn )T is the vector of input values and
y~n = (yn1 , yn2 , . . . , ynl )T is the vector of output values. Further, A,
B, C, D are matrices of appropriate dimensions which define the
system in question. The name of the model is derived from the concept of state variables which are parameters remembered (or rather,
capable of being remembered) by the system. One can find an evident parallel between this system and the functional diagrams of
finite automata since discrete systems are special cases of automata
with alphabets (input, output, state ones) composed of sets of values taken by signals ~x, ~y , ~z (in case of digital systems by respective
sets of admissible vectors of numbers) and transition functions and
output functions are expressed by the matrix relations of the model.
(2) Discrete linear transformation is a mapping {xn } → {xk } from the
original domain (in general a vector space CN1 , in most cases RN1 )
to the transformation domain (vector space CN2 or RN2 ) given by
(2.10)
xk =
NX
1 −1
n=0
ak,n xn ,
k = 0, . . . N2 − 1
4
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
or (in vector form) by X = A~x,
the form

a0,0
a0,1
 a1,0
a
1,1

A=
..
..

.
.
aN2 −1,0 aN2 −1,1
where A is an N2 × N1 matrix of
...
...
a0,N1 −1
a1,N1 −1
..
.





. . . aN2 −1,N1 −1
called the transformation kernel, elements of which are real or complex numbers. In mooost applications there is N1 = N2 and as a
result A is a square matrix. If it moreover regular, i.e. det(A) = 0,
the transformation is invertible, which means the there is a one-toone correspondence between the original sequence ~x and its transformation ~y called discrete spectrum. The inverse transformation
is ~x = A−1 ~y . Based on special properties of the matrix A we obtain respective special transforms such as Hadamar transform, Walsh
transform, discrete Fourier transform, Haar transform, etc. For details see e.g. [18].
(3) Inverse filtering and noised signal recovery. Very often, one needs to
recover an unknown signal from its garbled and noise influenced form
which is a result of passing through a garbling or noisy environment
(such as a communication channel). As an example recall one of
the recovery methods – Kalman filtering. Apart from the scalar
Kalman filter the vector filter is often used in applications since
the state vector need not contain the delayed values of one state
variable only but instead it can be compose of various variables that
have various physical meaning in the modelled system. This enables
a traightforward creation of well understandable models of physical
systems, the internal variable of which (i.e. the original signals)
can be Kalman filters estimated using external observation. Denote
vectors of delayed values of the original and observed signals by
T
~xn = (x1n , x2n , . . . , xR
n) ,
~yn = (yn1 , yn2 , . . . , ynR )T ,
where R is the order of the given model. In a similar way denote
the vectors of the driving noise and undesired noise ~gn and νn . The
state equation of the model describing the signal generation is
(2.11)
~xn+1 = An+1 ~xn + ~gn+1
while the output equation is
(2.12)
~yn+1 = Cn+1 ~xn + ~νn+1 .
In contrast to the standard form of the state description (2.9), we
have that it is the vector of undesired noise that is used in the output
equation of the system (instead of the input vector of the system)‘
otherwise there is obviously B = D = I from (2.9), i.e. the output
MULTI–ACTIONS OF VECTOR SPACES
5
equation of the system is
(2.13)
~yn+1 = An+1 Cn+1 ~xn + Cn+1~gn+1 + ~νn+1 .
In cases when we want to create signals of complicated structure
or model complicated systems (provided we choose sufficiently high
order R) we need to use vector and matrix mathematical apparatus
(e.g. in finite impluse response filters).
(4) Signal transfer. The state equation of a model describing generation
of a signal (2.11). Using the notation of item 3 we obtain the output
equation (2.12). Upon composition of the state and output equation
we get (2.13), i.e.
(2.14)
~yn+1 = An+1 Cn+1 ~xn + Cn+1~gn+1 + ~νn+1 .
Now denote by Mm,m (R) the set of square matrices of order m. One
can construct a Mealy-type automaton A = (Mm,m (R), X, Y, δ, σ),
where δ : Mm,m (R) × X → X, σ : Mm,m (R) × X → Y , δ(A, ~x) =
A~x + ~g , σ(C, ~u) = C~u + nu,
~ where ~g , ~ν are fixed column numerical vectors. Notice that if (Mm,m (R), ·) is the usual multiplicative
semigroup, then the defining condition of (finite) deterministic automata, sometimes – and in this paper – referred to as MAC, i.e.
mixed associativity condition, does not hold. However, it is easy
to construct an automaton satisfying this condition once its input
alphabet is a non-commutative semigroup or group.
In constructions of multiautomata motivated by the above mentioned
models included below we must assume that their numerical values are
greater than one as otherwise the necessary conditions would not hold. Further restrictions done below (e.g. for LA2 (T )) are motivated by the need to
obtain a quasi-ordered group.
It is very useful to generalize the theory signal processing to signals which
depend on more than one variable. By a multidimensional continuous signal
we mean a scalar function of a continuous vector argument ~x 7→ f (~x), where
the physical meaning of the vector components is not important. As a most
common example of a multidimensional signal one can take the static image as a brightness function of two planar coordinates f (x, y) or an image
changing in time as f (x, y, t), tomographic data f (x, y, z), etc. Notice that
tomography often uses 3D data changable in time, i.e. f (x, y, z, t). The
concept of multidimensional signals can be generalized also to vector signals ~x 7→ f (~x) such as in e.g. colour images, where a coulour image is a
two-dimensional signal with three-dimensional vector values, components of
which are the colour RGB coordinates. Apart from the algorithmic description of transformations it is very useful, especially in theoretic analysis, to
use matrix algebra techniques and special vectors and matrices.
6
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
3. Preliminaries
If not specified further on in the text we use only the very basic concepts of the hyperstructure theory such as hypergroupoid, semihypergroup,
hypergroup and join space. For their definitions cf. [20] or e.g. standard
books [13, 14]. Apart from these, we are going to use the following concepts and results included in [6]. By a quasi-ordered set G we mean a set
G with a reflexive and transitive binary relation R. By an extensive hypergroupoid (H, ∗) we mean a hypergroupoid (H, ∗) such that, for all a, b ∈ H,
there is {a, b} ⊆ a ∗ b. By an isotone mapping f we mean a mapping
f : (G, R) → (H, S) such that, for all x, y ∈ G, the fact that xRy implies f (x)Sf (y) while by a strongly isotone mapping f we mean a mapping
f : (G, R) → (H, S) such that f (R(a)) = S(f (a)) for any a ∈ G. By a homomorphism we mean a mapping f : (G, ∗G ) → (H, ∗H ) such that for any
pair of elements a, b ∈ G there is f (a ∗G b) ⊆ f (a) ∗H f (b). We speak of good
homomorphism if the equality holds instead of inclusion. For quasi-ordered
sets we define
(3.1)
a ∗R b = R(a) ∪ R(b) = R({a, b})
and dually
(3.2)
a ◦R b = R−1 (a) ∪ R−1 (b) = {x; xRa or xRb} = R−1 {a, b}.
Notice that in [14], p. 96, Proposition 2, condition (3.1) is proved to be
an equivalent condition for a hypergroupoid (H, ∗R ) to be a quasi-order
hypergroup, i.e. as defined first in [6], a hypergroup (H, ∗R ) such that, for
all a, b ∈ H, there is simultaneously a ∈ a3 ⊆ a2 and a ∗R b = a2 ∪ b2 . To
distinguish (3.1) and (3.2), we call the hypergroup (H, ∗R ) an upper order
hypergroup and the hypergroup (H, ◦R ) a lower order hypergroup. Moreover,
for any non–empty subset M ⊂ X we denote UR (M ), or LR (M ), the set of
all upper, or lower, bounds of the set M .
Lemma 3.1. ([6], Theorem 2.1, p. 150)
(1) Let (G, R) be a quasi–ordered set and ∗R defined using (3.1). Then
(G, ∗R ) is an extensive commutative hypergroup.
(2) Let (G, R), (H, S) be quasi–ordered sets, f : (G, R) → (H, S) a
(strongly) isotone mapping. Then f : (G, ∗R ) → (H, ∗S ) is a (good)
homomorphism of hypergroups (G, ∗R ), (H, ∗S ).
Lemma 3.2. ([6], Theorem 6.1., p. 182) Let (X, R) be an ordered set,
(X, ◦), (X, ∗) upper, or lower, order hypergroups determined by (X, R).
Then (X, ◦), or (X, ∗), is a join space if and only if for any pair of R–
incomparable elements a, b ∈ X such that the set LR (a, b), or UR (a, b), is
non-empty, the set UR (a, b), or LR (a, b), is also non–empty.
Remark 3.3. Generalization of the above results onto the case of quasiordered sets is straightforward.
MULTI–ACTIONS OF VECTOR SPACES
7
Further on, we will adjust some of the above general notation, especially
denote the (quasi–)ordering by “≤”. Also, for an arbitrary a ∈ X we will
by (a]≤ mean a set {y ∈ X; y ≤ a} and by [a)≤ a set {z ∈ X; a ≤ z}.
4. Hyperstructures motivated by modelling functions
In section 2 we have mentioned several specific modelling functions. Therefore, we now consider a commutative ring
(4.1)
RnF (Ω) = [F(Ω)]n = F(Ω) × . . . × F(Ω)
formed by n–dimensional vectors (f0 , . . . , fn−1 ) of real functions of m–variables
f : Ω → R, where ∅ 6= Ω ⊆ Rm , m ∈ N. The binary operation of addition is
defined as usually for vectors, i.e. component-wise, while multiplication is
defined by
(4.2)
(f0 , . . . , fn−1 ) · (g0 , . . . , gn−1 ) = (f0 g0 , . . . , fn−1 gn−1 ),
for any pair of vectors f~ = (f0 , . . . , fn−1 ), ~g = (g0 , . . . , gn−1 ) ∈ RnF (Ω).
Notice that (RnF (Ω), ·) is a commutative monoid. Therefore, we can consider
a suitable subset S ⊆ RnF (Ω) such that it is a submonoid of (RnF (Ω), ·).
On this ring RnF (Ω) and for this subset S we first define a binary relation
%S by the following rule:
If f~ = (f0 , . . . , fn−1 ), ~g = (g0 , . . . , gn−1 ) ∈ RnF (Ω), then we put f~%S~g whenever there exists a vector ~h = (h0 , . . . , hn−1 ) ∈ S such that
(4.3)
(g0 , . . . , gn−1 ) = (h0 , . . . , hn−1 ) · (f0 , . . . , fn−1 ) = (h0 f0 , . . . , hn−1 fn−1 ).
Next, we define a binary hyperoperation ∗d : RnF (Ω) × RnF (Ω) → P(RnF (Ω))
such that for any pair of vectors f~ = (f0 , . . . , fn−1 ), ~g = (g0 , . . . , gn−1 ) ∈
RnF (Ω) we put
(f0 , . . . , fn−1 ) ∗d (g0 , . . . , gn−1 ) =
{(f0 ϕ0 , . . . , fn−1 ϕn−1 ); (ϕ0 , . . . , ϕn−1 ) ∈ S}∪
∪ {(g0 ψ0 , . . . , gn−1 ψn−1 ); (ψ0 , . . . , ψn−1 ) ∈ S} = %S (f~) ∪ %S (~g ),
Notice that, in our notation “∗d ”, the symbol d suggests “direct” due to the
construction of the component-wise multiplication.
Theorem 4.1. The hypergroupoid (RnF (Ω), ∗d ) is an extensive commutative
hypergroup; it is a quasi–order hypergroup determined by the quasi–ordered
ring RnF (Ω). Moreover, (RnF (Ω), ∗d ) is a join space.
Proof. Since S is a submonoid of RnF (Ω), there is (1, . . . , 1) ∈ S. Thus the
relation %S is reflexive. It is also obviously transitive, i.e. it is a quasi–
ordering on the ring RnF (Ω). The fact, that (RnF (Ω), ∗d ) is an extensive
commutative hypergroup follows from Lemma 3.1.
8
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
The quasi–order hypergroup (RnF (Ω), ∗d ) is determined by the above defined binary relation %S ⊂ RnF (Ω) × RnF (Ω), which is a quasi–ordering on
the ring RnF (Ω). Suppose f~, ~g ∈ RnF (Ω) are vectors such that L%S (f~, ~g ) 6= 0,
which means that there exists a vector ~h ∈ RnF (Ω) such that ~h%S f~, ~h%S~g .
Then for a suitable pair of vectors ~v , ~u ∈ S ⊆ RnF (Ω) we have ~h · ~u =
f~, ~h · ~v = ~g . Then f~ · ~v = ~h · ~u · ~v = ~h · ~v · ~u = ~g · ~u. Denoting p~ = f~ · ~v = ~g · ~u
we have that f~%S p~, ~g %S p~ thus U%S (f~, ~g ) 6= 0. Consequently, by Lemma 3.2
the hypergroup (RnF (Ω), ∗d ) is a join space.
4.1. Binary hyperoperation derived from the direct product of n–
dimensional linear spaces of functions. For a positive integer n ≥ 2
and an interval T ⊆ R denote by LAn (T ) the set of all linear differential
operators of the n–th order L(p0 , . . . , pn−1 ), where pk ∈ C(T ), with the
action
(4.4)
L(p0 , . . . , pn−1 )y = y (n) (x) + pn−1 (x)y n−1 (x) + ... + p0 (x)y(x)
for any y ∈ C n (T ). Denote by [C(T )]n = C(T ) × ... × C(T ) the linear space
|
{z
}
n
of vectors p~ = (p0 , . . . , pn−1 ) of continuous functions. Similarly as above –
yet now on the set LAn (T ) – we define hyperoperation “∗d ” by the rule:
L(p0 , . . . , pn−1 ) ∗d L(g0 , . . . , gn−1 ) =
{L(ϕ0 p0 , . . . , ϕn−1 pn−1 ); (ϕ0 , . . . , ϕn−1 ) ∈ [C(T )]n }∪
∪ {L(ψ0 g0 , . . . , ψn−1 gn−1 ); (ψ0 , . . . , ψn−1 ) ∈ [C(T )]n },
for any pair of linear differential operators L(p0 , . . . , pn−1 ), L(g0 , . . . , gn−1 ) ∈
LAn (T ). For easier reading and manipulation we will use also the following
notation:
(4.5)
{L(ϕ0 p0 , . . . , ϕn−1 pn−1 ); (ϕ0 , . . . , ϕn−1 ) ∈ [C(T )]n } = L (~
p)
and similarly for L (~g ), which enables us to write the definition of “∗d ” on
LAn (T ) in a concise form L(p0 , . . . , pn−1 ) ∗d L(g0 , . . . , gn−1 ) = L (~
p) ∪ L (~g ),
i.e.
(4.6)
L(~
p) ∗d L(~g ) = L (~
p) ∪ L (~g ).
Theorem 4.2. The hypergroupoid (LAn (T ), ∗d ) is a join space.
Proof. Follows immediately from Theorem 4.1 and analogy of its proof. Indeed, the set L (~
p) is the principal upper set of (LAn (T ), ≤) generated by the
operator L(p0 , . . . , pn−1 ), where the relation ≤ is a quasi-ordering defined
by
L(p0 , . . . , pn−1 ) ≤ L(q0 , . . . , qn−1 )
whenever there exists a vector (ϕ0 , . . . , ϕn−1 ) ∈ [C(T )]n such that qk = pk ϕk ,
for all k = 0, . . . , n − 1. Now, similarly as in the proof of Theorem 4.1, we
easily obtain the quasi-ordered set which satisfies the criterion formulated
in Lemma 3.2. Suppose that there exists L(~
ϕ) ∈ (L(~
p)]≤ ∪ (L(~q)]≤ for an
MULTI–ACTIONS OF VECTOR SPACES
9
arbitrary pair of operators L(~
p), L(~q) ∈ LAn (T ), i.e. p~ = ϕ
~ · ~u, ~q = ϕ
~ · ~v for
a suitable pair of vectors ~u, ~v ∈ [C(T )]n . Then
p~ · ~v = ϕ
~ · ~u · ~v = ~u · ϕ
~ · ~v = ~u · ~q = ~q · ~u.
Denoting w
~ = p~ · ~v = ~q · ~u, we have L(~
p) ≤ L(w),
~ L(~q) ≤ L(w),
~ i.e.
L(w)
~ ∈ [L(~
p))≤ ∩ [L(~q))≤ and the condition is satisfied.
Remark 4.3. It is important to notice that when considering the case of
(LAn (T ), ∗d ), where n = 2, we get the modelling functions and corresponding differential equations discussed in section 2.
In section 5 of this paper we are going to construct quasi–multiautomata
with the input hypergroup (Rn , •), where Rn is the n–dimensional vector
space over the field of real numbers, i.e.
(4.7)
Rn = {(s0 , . . . , sn−1 ); sk ∈ R, k = 0, ..., n − 1},
on which we define a binary hyperoperation “•” by
(4.8)
(r0 , . . . , rn−1 ) • (s0 , . . . , sn−1 ) = {(t0 , . . . , tn−1 ), tk ≥ rk sk , k = 0, ..., n − 1}.
Lemma 4.4. The hypergroupoid (Rn , •) is a commutative hypergroup.
Proof. Suppose arbitrary (r0 , . . . , rn−1 ) = ~r, (s0 ,S. . . , sn−1 ) = ~s, (t0 , . . . , tn−1 ) =
~t ∈ Rn . Then ~r • (~s • ~t) = ~r • {~u, ~u ≥ ~s~t} = ~u≥~s~t ~r~u = {~v ; ~v ≥ ~r~s~t} =
S
rw
~ = (~r • ~s) • ~t. Thus (Rn , •) satisfies the associativity axiom and is
w≥~
~ r~s ~
a semihypergroup.
Next, we verify that the semihypergroup (Rn , •) satisfies the reproduction
axiom, i.e. ~r • Rn = Rn for all ~r ∈ R. It is evident that ~r • Rn ⊆ Rn for
any ~r ∈ Rn . Consider arbitrary element ~t ∈ Rn . Then there exists the
element ~s ∈ Rn such that ~t ∈ ~r • ~s; i.e. tk ≥ rk sk , k = 0, ..., n − 1. Define
the coefficients sk = rtkk for any k ∈ {0, ..., n − 1}. Then rk · rtkk = tk for
any k = 0, ..., n − 1, thus ~t ∈ ~r • ~s. Consequently Rn ⊆ ~r • Rn and we have
~r • Rn = Rn . Thus the semihypergroup (Rn , •) is a hypergroup.
4.2. A group of second–order linear differential operators of the
almost–Jacobi form (shortly al–JF). In this subsection, for practical
computational purposes, we exclude zero from the time interval, i.e. we
regard an interval T = hε, ∞), where ε is arbitrarily small.
If we generalize equation (2.6) or multiply equation (2.8) so that there
are no fractions, we get
(4.9)
[1 − exp(−ct)]2 y 00 (t) + [bc2 · exp(−ct)] · [b exp(−ct) − 1]y(t) = 0,
where [1 − exp(−ct)]2 is obviously nonegative for all x ∈ R. This makes us
study these operators in general.
Denote by al-JA2 (T ) the set of differential operators J(p, q) : C 2 (T ) →
C(T ) defined by
(4.10)
J(p, q)y = p(x)y 00 (x) + q(x)y(x), x ∈ T,
10
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
i.e.
(4.11)
al-JA2 (T ) = {J(p, q); p, q ∈ C(T ), p(x) 6= 0, x ∈ T = hε, ∞)}
with ε > 0 arbitrarily small. When we define on the set al-JA2 (T ) a binary
operation by
(4.12)
J(p1 , q 1 ) · J(p2 , q 2 ) = J(p1 p2 , p1 q2 + q1 ),
we obtain a non-commutative group with the unit element J(1, 0) and
J −1 (p, q) = J( p1 , − pq ) for an arbitrary J(p, q) ∈ al-JA2 (T ). Notice that
the above defined multiplication (4.12) is derived from the multiplication on
LA2 (T ) = {L(p, q); p, q ∈ C(T ), p(x) 6= 0, x ∈ T = hε, ∞)}, where
(4.13)
L(p, q)y = y 00 + p(x)y 0 + q(x)y
and
(4.14)
L(p1 , q1 ) · L(p2 , q2 ) = L(p1 p2 , p1 q2 + q1 ).
The name “almost Jacobi form” relates to the fact that operators (4.10) are
similar to the Jacobi form of differential equations (2.6) and (2.8) yet differ
in the leading coefficient. The set of such operators J(0, p) was introduced
in [11], where it was denoted JA2 (T ) with the aritmetics of the operators
defined by
(4.15)
J(0, p1 ) · J(0, p2 ) = J(0, p1 p2 ).
By J+ A2 (T ) we denote the subset of JA2 (T ) which consists of operators
with p positive for all x ∈ T , i.e.
(4.16)
J+ A2 (T ) = {J(0, p); p ∈ C + (T ), p(x) > 0 for all x ∈ T }.
Theorem 4.5. Let T = hε, ∞), ε > 0. Then the mapping F : al-JA2 (T ) →
LA2 (T ) defined by F (J(p, q)) = L(p, q) for any operator J(p, q) ∈ al-JA2 (T )
is an isomorphism of the group (al-JA2 (T ), ·) onto the group (LA2 (T ), ·).
Moreover
(J+ A2 (T ), ·) / (al-JA2 (T ), ·) ∼
= (LA2 (T ), ·),
i.e. the group (J+ A2 (T ), ·) is a normal subgroup of (LA2 (T ), ·), which is
isomorphic to (al-JA2 (T ), ·).
Proof. Define F : al-JA2 (T ) → LA2 (T ) by F (J(p, q)) = L(p, q) for any operator J(p, q) ∈ al-JA2 (T ). The mapping F is evidently surjective. We show
that it is also injective. Indeed, suppose F (J(p1 , q 1 )) = F (J(p2 , q 2 )) for some
pair of operators J(p1 , q 1 ), J(p2 , q 2 ) ∈ al-JA2 (T ). Then p1 (x)y 00 + q1 (x)y =
p2 (x)y 00 + q2 (x)y for any function y ∈ C 2 (T ). Considering the function
y(x) ≡ 1 we obtain q1 = q2 , thus p1 (x)y 00 = p2 (x)y 00 for any y ∈ C 2 (T ).
For y(x) = 21 x2 we obtain p1 = p2 (on the whole interval T), consequently
J(p1 , q 1 ) = J(p2 , q 2 ). Hence, the mapping F : al-JA2 (T ) → LA2 (T ) is bijective. Further, for an arbitrary pair of operators J(p1 , q 1 ), J(p2 , q 2 ) ∈
MULTI–ACTIONS OF VECTOR SPACES
11
al-JA2 (T ) we have
F (J(p1 , q 1 ) · J(p2 , q 2 )) = F (J(p1 · p2 , p1 q2 + q1 )) =
= L(p1 · p2 , p1 q2 + q1 ) = L(p1 , q 1 ) · L(p2 , q 2 ) = F (J(p1 , q 1 )) · F (J(p2 , q 2 )),
thus F : (al-JA2 (T ), ·) → (LA2 (T ), ·) is an isomorphism.
Consider now a mapping G : J+ A2 (T ) → L11 A2 (T ) defined for all operators J(0, p) ∈ J+ A2 (T ) by G(J(0, p)) = L(1, ln p). There is
G(J(0, p) · J(0, q)) = G(J(0, pq)) = L(1, ln (pq)) = L(1, ln p + ln q) =
L(1, ln p) · L(1, ln q) = G(J(0, p)) · G(J(0, q)).
Moreover, for an arbitrary functions q ∈ C(T ) and p ∈ C + (T ) such that
p(x) = eq(x) > 0 there is G(J(0, p)) = L(1, ln p) = L(1, q). Thus, altogether
we have that (J+ A2 (T ), ·) is isomorphic to (L11 A2 (T ), ·). And if we realize
that (L11 A2 (T ), ·) is a normal subgroup of (LA2 (T ), ·) – notice that the
inverse element of an arbitrary element L(p, q) ∈ LA2 (T ) is L( p1 , − pq ) – we
obtain the theorem.
5. Products of quasi-multiautomata
Numerous types of automata exist and have been studied. By an automaton without output, i.e. deterministic automaton without output alphabet, we
mean a triad A = (A, S, δ), where the non-empty set A is called the set of
input symbols, the non-empty set S is called the state set and δ : A × S → S
is a mapping, known as transition map, such that, if (A, ·) is a monoid, there
is
(1) δ(e, s) = s for any s ∈ S and the identity e ∈ A,
(2) δ(b, δ(a, s)) = δ(a · b, s) for all a, b ∈ A and s ∈ S.
Though mostly nameless, we refer to it as MAC, i.e. Mixed Associativity Condition, a name sometimes used. Now we give two constructions of
automata, the transition map of which fulfill the MAC condition.
Example 5.1. Denote G the additive group of all vectors ~g . Further, let
F = {[A, ~g ]; A ∈ Mm,m (R), ~g ∈ G}. Define a binary operation on F by
(5.1)
[A, ~g ] · [B, ~q] = [A · B, A~q + ~g ].
Obviously, (F, ·) is a non-commutative semigroup, or a non-commutative
group if Mm,m (R) is a multiplicative group of regular matrices. Further,
define a map δ : F × X by
(5.2)
δ([A, ~g ], ~x) = A~x + ~g
for every [A, ~g ] ∈ F and ~x ∈ X. Then for every quadruple [A, ~g ], [B, ~q] ∈ F
and ~x ∈ X there is
δ([A, ~g ], δ([B, ~q], ~x)) = δ([A, ~g ], B~x + ~q) = A(B~x + ~q) + ~g = AB~x + A~q + ~g =
= δ([AB, A~q + ~g ], ~x) = δ([A, ~g ] · [B, ~q], ~x),
which is exactly MAC for the transition map δ.
12
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
Of course, the transition map may be defined in other ways as well. In the
following example we make use of the idea of direct, i.e. component-wise,
product of vectors.
Example 5.2. First, for all f~, ~g ∈ V , where f~ = (f1 , . . . , fm ), ~g = (g1 , . . . , gm )
put
(5.3)
f~ ~g = ~h = (f1 g1 , . . . , fm gm ),
where (fi gi )(x) = max{fi (x), gi (x)}, x ∈ R, i ∈ {1, 2, . . . , m}. This
binary operation “” on a vector space V is idempotent, associative and
commutative, i.e. (V, ) is semilattice. Further, define action δ : Mm,m (R)×
X → X by
(5.4)
δ(A, ~x) = A~xT ~g T
where ~x ∈ X and ~gδT is a fixed vector chosen for δ. For square matrices
A, B ∈ Mm,m (R)and ~x ∈ X we have
δ(B, δ(A, ~x)) = δ(B, (A~xT ~gδT )) = BA~xT ~gδT ~gδT = BA~xT ~gδT = δ(BA, x̃T ).
Therefore, the MAC condition holds.
In the following definition we recall a hyperstructure generalization of
a quasi-automaton called a quasi-multiautomaton (without output). This
concept has been used in e.g. [7, 8, 9, 10, 12]; for a use similar to the
context of our paper see [7]. Condition (5.5) is called GMAC, which stands
for “Generalised Mixed Associativity Condition”.
Definition 5.3. A quasi–multiautomaton (without output) is a triad A =
(H, S, δ), where (H, ∗) is a semihypergroup, S is a non–empty set and δ : H×
S → S is a transition map satisfying the condition:
(5.5)
δ(b, δ(a, s)) ∈ δ(a ∗ b, s) for all a, b ∈ H, s ∈ S.
The set S is called the state–set of the quasi–multiautomaton A, the hyperstructure (H, ∗) is called the input semihypergroup of the quasi–multiautomaton
A and δ is called the transition function. Elements of the set S are called
states, elements of the set H are called input symbols (or words).
Dörfler [15] suggested several types of products of automata – a homogeneous product, a heterogeneous product (and also a Cartesian composition
which is not discussed in this paper). The following Dörfler’s [15] definition
is included with formal changes such as swapping the order of the input-set
and the state-set or using our notation.
Definition 5.4. [15] Let A1 = (H, S1 , δ1 ), A2 = (H, S2 , δ2 ) and A3 =
(M, S3 , δ3 ) be automata. By the homogeneous product A1 × A2 we call
the automaton (H, S1 × S2 , δ1 × δ2 ), where δ1 × δ2 : H × (S1 × S2 ) → S1 × S2
is a mapping satisfying
(δ1 × δ2 )(h, (s, t)) = (δ1 (h, s), δ2 (h, t)); s ∈ S1 , t ∈ S2 , h ∈ H,
MULTI–ACTIONS OF VECTOR SPACES
13
while the heterogeneous product A1 ⊗ A3 is the automaton (H × M, S1 ×
S3 , δ1 ⊗ δ3 ), where δ1 ⊗ δ3 : (S1 × S3 ) × (H × M ) → S1 × S3 is a mapping
satisfying
δ1 ⊗δ3 ((h, m), (s, t)) = (δ1 (h, s), δ3 (m, t)) for all h ∈ H, m ∈ M, s ∈ S1 , t ∈ S3 .
In this section we generalize the concepts of homogeneous and heterogeneous products as we study them not on quasi–automata but on quasi–
multiautomata with the input sets being hyperstructures discussed in the
above sections and later on in Theorem 5.9.
5.1. Homogeneous product.
Theorem 5.5. The structure A1 = ((Rn , •), LAn (T ), δ1 ), where δ1 : Rn ×
LAn (T ) → LAn (T ) is defined by
δ1 (~r, L(~
p)) = (~r · L(~
p)) = L(r0 p0 , . . . , rn−1 pn−1 ),
is a quasi–multiautomaton.
Proof. Thanks to Lemma 4.4 we already know that (Rn , •) is a hypergroup.
Next, we show that the structure A1 satisfies GMAC (5.5), i.e.
δ1 (~r, δ1 (~s, L(~
p))) ∈ δ1 (~r • ~s, L(~
p)).
The calculation of the left hand side:
δ1 (~r, δ1 (~s, L(~
p))) = δ1 (~r, L(s0 p0 , . . . , sn−1 pn−1 ))
= L(r0 s0 p0 , . . . , rn−1 sn−1 pn−1 ).
The calculation of the right hand side:


[
δ1 (~r • ~s, L(~
p)) = δ1 
(~t, L(~
p)) = {L(t0 p0 , . . . , tn−1 pn−1 ); ~t ∈ Rn , ~t ≥ ~r~s}.
~t≥~
r~s
For ~t = ~r~s we have L(r0 s0 p0 , . . . , rn−1 sn−1 pn−1 ) = L(t0 p0 , . . . , tn−1 pn−1 ); ~t ∈
Rn . Thus L(r0 s0 p0 , . . . , rn−1 sn−1 pn−1 ) ∈ δ1 (~r • ~s, L(~
p)) and the structure
A1 = ((Rn , •), LAn (T ), δ1 ) is a quasi–multiautomaton.
Theorem 5.6. The structure A2 = ((Rn , •), RnF (Ω), δ2 ), where δ2 : Rn ×
RnF (Ω) → RnF (Ω) is defined by
δ2 (~r, p~) = ~r · p~ = (r0 p0 , . . . , rn−1 pn−1 ),
is a quasi–multiautomaton.
Proof. The proof is analogous to the proof of Theorem 5.5.
Since the input hypergroups of quasi–multiautomata A1 and A2 are the
same, we can construct their homogeneous product as an analogy to the
Definition 5.4.
14
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
Theorem 5.7. Consider two quasi–multiautomata, A1 = ((Rn , •), LAn (T ), δ1 )
and A2 = ((Rn , •), RnF (Ω), δ2 ), and define
Ahom = ((Rn , •), LAn (T ) × RnF (Ω), δ1 × δ2 ),
(5.6)
where
(δ1 × δ2 )(~r, (L(~
p), p~)) = (δ1 (~r, L(~
p)), δ2 (~r, p~))
for all L(~
p) ∈ LAn (T ), p~ ∈ RnF (Ω) and ~r ∈ Rn . Then Ahom is a quasi–
multiautomaton.
Proof. We must verify GMAC (5.5), i.e.
(δ1 × δ2 )(~r, (δ1 × δ2 )(~s, (L(~
p), ~q))) ∈ (δ1 × δ2 )(~r • ~s, (L(~
p), ~q)),
in Ahom . There is
(δ1 × δ2 )(~r, (δ1 (~s, L(~
p)), δ2 (~s, ~q))) =
(δ1 × δ2 )(~r, (L(s0 p0 , . . . , sn−1 pn−1 )), (s0 q0 , . . . , sn−1 qn−1 ))
(δ1 (~r, L(~sp~)), δ2 (~r, ~s~q)) = (L(~r~sp~), ~r~s~q) ∈
∈ {(L(~tp~), ~t~q); t ∈ {(u0 , . . . , un−1 ), uk ≥ rk sk , k = 0, . . . n − 1}}
with respect to ~t = (r0 s0 , . . . , rn−1 sn−1 ). Yet
{(L(~tp~), ~t~q); t ∈ {(u0 , . . . , un−1 ), uk ≥ rk sk , k = 0, . . . n − 1}} =
= {((δ1 (~t, L(~
p))), δ2 (~t~q)); t ∈ {(u0 , . . . , un−1 ), uk ≥ rk sk , k = 0, . . . n − 1}} =
= {(δ1 (~t, L(~
p)), δ2 (~t, ~q)); ~t ∈ ~r • ~s} = (δ1 × δ2 )(~r • ~s, (L(~
p), ~q)),
thus the structure Ahom is a quasi–multiautomaton.
5.2. Heterogeneous product. Next, we proceed to the concept of the
heterogeneous product of quasi-multiautomata. Unlike with homogeneous
products we now need two different input hyperstructures. For this purpose
the set of square matrices Mn,n (R+ ) of order n with non-negative real entries
seems suitable. On Mn,n (R+ ) regard such a relation ≤ that for matrices A =
(aij ), B = (bij ) we write A ≤ B whenever aij ≤ bij for all i, j ∈ {1, . . . , n}.
Further on we denote Nn = {1, . . . , n}.
We define the hyperoperation “◦” on Mn,n (R+ ) by:
A ◦ B = {C ∈ Mn,n (R+ ); min{A, B} ≤ C},
(5.7)
i.e.


 c11
. . .

cn1
 

a11 . . . a1n
b11 . . . b1n
 . . . . . . . . .  ◦ . . . . . . . . .  =
an1 . . . ann
bn1 . . . bnn


. . . c1n

. . . . . .  ; min{aij , bij } ≤ cij , i, j ∈ {1, . . . , n} ,

. . . cnn
where aij , bij , cij ∈ R.
MULTI–ACTIONS OF VECTOR SPACES
15
Remark 5.8. Notice that the above defined minimum of square matrices is an
associative and commutative operation on Mn,n (R+ ). Moreover, the above
defined ordering of matrices is reflexive and transitive. Finally, (Mn,n (R+ ), min, ≤
) is obviously a quasi-ordered semigroup. Thus, (Mn,n (R+ ), ◦) is a commutative EL–semihypergroup in the sense of [24, 25]. However, since (Mn,n (R+ ), min)
is not a group, the fact that (Mn,n (R+ ), ◦) is a hypergroup must be proved
using other means. Also notice that if we aimed at constructing the Cartesian composition of quasi-multiautomata, the fact that structures (Mn,n (R+ ), ◦)
and (Rn , •) are disjoint for n ≥ 2 would become very important.
Theorem 5.9. The hypergroupoid (Mn,n (R+ ), ◦) is a commutative hypergroup.
Proof. Suppose A, B, C ∈ Mn,n (R+ ). Thanks to Remark 5.8 we already
know that (Mn,n (R+ ), ◦) is a commutative semihypergroup, i.e. we need to
check the validity of the reproduction axiom only. Since it is evident that
A ◦ Mn,n (R+ ) ⊆ Mn,n (R+ ), for any A ∈ Mn,n (R+ ), we have to show that
for all A ∈ Mn,n (R+ ) there is Mn,n (R+ ) ⊆ A ◦ Mn,n (R+ ).
For a fixed A ∈ Mn,n (R+ ) and an arbitrary M ∈ Mn,n (R+ ) we define
a matrix B ∈ Mn,n (R+ ) such that for its entries bij , i, j ∈ Nn , we put
bij = min{aij , mij }. Thanks to the assumption of real entries such a matrix
can always be constructed and there is B = min{A, M} and B ≤ M as well
as B ≤ A. Since min{A, B} = B ≤ M, there is
M ∈ A ◦ B = {C ∈ Mn,n (R+ ); min{A, B} ≤ C},
i.e. M ∈ M ◦ Mn,n (R+ ). Thus, Mn,n (R+ ) ⊆ A ◦ Mn,n (R+ ), which means
that (Mn,n (R+ ), ◦) is a commutative hypergroup.
The following theorem gives a construction of a quasi-multiautomaton
with the input semihypergroup of square matrices and the state set RnF (Ω).
In order to secure validity of the GMAC condition (5.5), we will assume
that the non-negative real entries of matrices M ∈ Mn,n (R+ ) are such that
mij ≥ 1. We will denote such a set as Mn,n (R+
1 ). Notice that this restriction
has no influence on validity of Theorem 5.9 as – due to the choice of operation
min – we could in fact restrict the value of mij by an arbitrary real number
instead of 1.
F
Theorem 5.10. The structure A3 = ((Mn,n (R+
1 ), ◦), Rn (Ω), δ3 ), where δ3 :
+
F
F
(Mn,n (R1 ), ◦) × Rn (Ω) → Rn (Ω) is defined by δ3 (A, p~) = p~ · A = ~q, where


a11 . . . a1n
p~ · A = (p0 , . . . , pn−1 ) ·  . . . . . . . . .  = (q0 , . . . , qn−1 ) = ϕ
~,
an1 . . . ann
is a quasi–multiautomaton.
16
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
Proof. We show that GMAC (5.5), i.e. δ3 (B, δ3 (A, p~)) ∈ δ3 (A ◦ B, p~), holds
in A3 . The calculation of the left-hand side:



a11 . . . a1n
δ3 ((bij ), δ3 ((aij ), (p0 , . . . , pn−1 ))) = δ3 (bij ), (p0 , . . . , pn−1 ) ·  . . . . . . . . .  =
an1 . . . ann
!!
n
n
X
X
= δ3 (bij ),
ai1 pi−1 , . . . ,
=
ain pi−1
i=1
n
X
=
ai1 pi−1 , . . . ,
i=1
=
b11
n
X
i=1
n
X
! 
ain pi−1
i=1
n
X
ai1 pi−1 + . . . + bn1
i=1
b11 . . .
· . . . . . .
bn1 . . .
ain pi−1 , . . . ,
i=1
b1n
n
X
ai1 pi−1 + . . . + bnn
i=1

=

b1n
. . . =
bnn
n
X
aij bj1 pj−1 , . . . ,
i,j=1
n
X
n
X
!
ain pi−1
=
i=1

aij bjn pj−1  = (q0 , . . . , qn−1 ).
i,j=1
The calculation of the right-hand side:
δ3 (A ◦ B, (p0 , . . . , pn−1 )) =
= {(p0 , . . . , pn−1 ) · C; C = (cij ); cij ≥ min{aij , bij }, i, j ∈ Nn } =
= {(c11 p0 ) + . . . + cn1 pn−1 , . . . ,
c1n p0 + . . . + cnn pn−1 ); cij ≥ min{aij , bij }, i, j ∈ Nn }.
+
Since Mn,n (R+
1 ) = {M ∈ Mn,n (R ); M = (mij ), mij ≥ 1, i, j ∈ Nn }, we
have in general qk−1 = c1k p0 + . . . + cnk pn−1 . Since we have
n
X
cjk =
aik bjk ≥ min{aik bjk } ≥ 1,
j,i=1
we obtain

(q0 , . . . , qn−1 ) = 
n
X
aij bj1 pj−1 , . . . ,
i,j=1
n
X

aij bjn pj−1  ∈
i,j=1
∈ {(c11 p0 + . . . + cn1 pn−1 , . . . ,
c1n p0 + . . . + cnn pn−1 ); cij ≥ min{aij , bij }, i, j ∈ Nn } =
MULTI–ACTIONS OF VECTOR SPACES
17
= δ3 (A ◦ B, (p0 , . . . , pn−1 )).
Hence GMAC (5.5) is satisfied.
+
Example 5.11. The restriction to use Mn,n (R+
1 ) instead of Mn,n (R ) in
Theorem 5.10 is essential. E.g. for matrices
0.3 0.5
0.2 0.5
A=
, B=
0.4 0.3
0.1 0.1
and a vector of constant functions p~ = (p1 , p2 ) such as e.g. p~ = (1, 1) there
is
δ3 (B, δ3 (A, p~)) 6∈ δ3 (A ◦ B, p~).
Indeed, the calculation of the left-hand side of the GMAC condition (5.5)
gives δ3 (A, p~) = p~ · A = (0.7, 0.8). Then δ3 (B, (0.7, 0.8)) = (0.7, 0.8) ·
B = (0.21, 0.43). For the calculation of the right-hand side of the GMAC
condition (5.5) first consider that min{A, B} = B. Now, p~ · B = (0.3, 0.6)
and since the right-hand side is δ3 (A ◦ B, p~) = {(q0 , q1 ); q0 ≥ 0.3, q1 ≥ 0.6},
we see that the condition does not hold for this particular choice of A, B
and p~.
Finally, we construct the heterogeneous product of multiautomata A1 and
A3 constructed in Theorem 5.5 and Theorem 5.10.
Theorem 5.12. Consider two quasi–multiautomata, A1 = ((Rn , •), LAn (T ), δ1 )
F
and A3 = ((Mn,n (R+
1 ), ◦), Rn (Ω), δ3 ), and define
F
Ahet = ((Rn , •) × (Mn,n (R+
1 ), ◦), LAn (T ) × Rn (Ω), δ1 ⊗ δ3 ),
where
δ1 ⊗ δ3 ((~r, A), (L(~
p), ~q)) =
= (δ1 ((r0 , . . . , rn−1 ), (p0 , . . . , pn−1 )), δ3 ((aij ), (q0 , . . . , qn−1 ))) =
!!
n
n
X
X
= L(~rp~),
ai1 qi1 , . . . ,
ain qi1
,
i=1
i=1
for all L(~
p) ∈ LAn (T ), ~q ∈ RnF (Ω), A ∈ Mn,n (R+
r ∈ Rn . Then Ahet is
1 ) and ~
a quasi–multiautomaton.
Proof. We must verify GMAC (5.5) in Ahet , i.e.:
(δ1 ⊗ δ3 ) ((~s, B), (δ1 ⊗ δ3 )((~r, A), (L(~
p), p~))) ∈
∈ (δ1 ⊗ δ3 ) ((~r, A) (~s, B), (L(~
p), ~q)) =
(δ1 ⊗ δ3 )((~r • ~s, A ◦ B), (L(~
p), ~q))
18
J. CHVALINA, Š. KŘEHLÍK, AND M. NOVÁK
The left-hand side:
(δ1 ⊗ δ3 ) ((~s, B), (δ1 ⊗ δ3 )((~r, A), (L(~
p), p~))) =
= (δ1 ⊗ δ3 ) (~s, B), L(~rp~),
n
X
ai1 qi−1 , . . . ,
i=1

L(~s~rp~),
!!!
ain qi−1
n
X

b1n
. . .  =
bnn
n
X
ai1 b11 qi−1 + . . . +
i=1
n
X
=
i=1
! 
b11 . . .

ai1 qi−1 , . . . ,
ain qi−1 · . . . . . .
bn1 . . .
i=1
i=1
n
X
L(~s~rp~)),
=
n
X
ain bn1 qi−1 , . . . ,
i=1
n
X
ai1 b1n qi−1 + . . . +
i=1
n
X
!!
ain bnn qi−1
.
i=1
The right-hand side:
(δ1 ⊗ δ3 )((~r • ~s, A ◦ B), (L(~
p), ~q)) =
= (δ1 ⊗ δ3 )
(
c11 . . .
. . . . . .
cn1 . . .
=


δ
 3
{(t0 , . . . , tn−1 ); tk ≥ rk sk , k = 0, 1, . . . , n − 1},

)!
!
c1n
. . .  ; cij ≥ min{aij , bij }, i, j ∈ Nn
, (L(~
p), ~q) =
cnn
{δ1 ((t0 , . . . , tn−1 ), L(~
p)); tk ≥ rk sk , k = 0, 1, . . . , n − 1} ,

c11 . . .
. . . . . .
cn1 . . .
=
!
 !
c1n

. . .  , ~q ; cij ≥ min{aij , bij }, i, j ∈ Nn
=

cnn
{L(~tp~); tk ≥ rk sk , k = 0, 1, . . . , n − 1},

c11 . . .
(q0 , . . . , qn−1 ) . . . . . .
cn1 . . .
(

)!
c1n
. . .  ; cij ≥ min{aij , bij }, i, j ∈ Nn
=
cnn
= ({L(~tp~); tk ≥ rk sk , k = 0, 1, . . . , n − 1},
{(c11 q0 + . . . + cn1 qn−1 , . . . , c1n q0 + . . . + cnn qn−1 );
cij ≥ min{aij , bij }, i, j ∈ Nn }).
MULTI–ACTIONS OF VECTOR SPACES
19
Similarly as in the proof of Theorem 5.10 we for ~t = (t0 , . . . , tn−1 ) =
(r0 , . . . , rn−1 ) · (s0 , . . . , sn−1 ) = ~r · ~s and cij = aij bij , for i, j ∈ Nn , obtain
(δ1 ⊗δ3 ) ((~s, B), (δ1 ⊗ δ3 )((~r, A), (L(~
p), p~))) ∈ (δ1 ⊗δ3 )((~r•~s, A◦B), (L(~
p), ~q)).
Consequently, GMAC (5.5) is satisfied in Ahet .
Acknowledgement
The second author was supported by the Norway Grants under Grant ”Mathematical Education Through Modelling Authentic Situations - METMAS”,
registration number NF-CZ07-ICP-3-201-2015.
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(Jan Chvalina) Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00
Brno, Czech Republic
E-mail address: [email protected]
(Štěpán Křehlı́k) Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8,
616 00 Brno, Czech Republic
E-mail address: [email protected]
(Michal Novák) Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8,
616 00 Brno, Czech Republic
E-mail address: [email protected]