Abstract
Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers.
1. Introduction
Integer semigroups have been studied in different contexts (cf. [1–4]). One of the well known binary operations is the Fibonacci multiplication, introduced by Knuth [1]. This operation is related to the dynamical system obtained by the Fibonacci substitution (cf. [3]). It has been generalized in different ways to the tribonacci substitution; see [4–6]. The author studied the usage of binary operations on the set of nonnegative integers, in order to describe the self-similar structure of the -bonacci substitutions [4] and on the so-called flipped tribonacci substitution [7].
In the present article, we introduce binary operations on infinite subsets of the nonnegative integers so that they are integer semigroups. These binary operations are associated with a substitution, in particular to the numeration system defined by the substitution, the so-called Dumont-Thomas numeration system. These numeration systems play an important role in the study of Rauzy fractals; for details see [8, 9] and references within. The integer semigroups defined in the paper, in general, do not have identity elements. We explore when they have identity elements. In Theorem 3, we show that these semigroups are not finitely generated.
Moreover, we show that these semigroups define actions on the set of positive integers, and we describe the orbits of these actions.
In Section 4, we study the density of these semigroups as subsets of the positive integers.
2. Substitutions and Automata
A substitution on a finite alphabet is a map from to the set of finite words on ; that is, . This map extends to by concatenation by and , for all , . Let denote the set of one-sided infinite sequences in . The map is extended to in the obvious way. We call a fixed point of if and periodic if there exists so that it is fixed for . There is a dynamical system associate to the sequence , called substitution dynamical system; for details see [8–10].
The incidence matrix of the substitution is defined as the matrix whose entry is the number of occurrences of the symbol in the word , for . We say the substitution is primitive if its incidence matrix is primitive; that is, all the entries of are positive for some . We shall assume throughout the paper that the substitutions are primitive.
For a primitive substitution there are a finite number of periodic points. So, we can assume without loss of generality that a primitive substitution has a fixed point; say and ; that is, the first symbol of is .
Let be a set of finite words on the alphabet . An automaton over , is a direct graph labelled by elements of . is the set of states, is the set of initial states, is the set of labels, and is the set of labelled edges or transitions. If , we say that is a transition between and .
The prefix automaton of the substitution (cf. [11]) is the automaton so that one has the following.(i).(ii), the set of proper prefixes of the words , for . We will denote by the empty prefix.(iii) is in if , are elements of , and is a prefix of .(iv).
In Figure 1, it is shown the prefix automaton of the -bonacci substitution; that is, , and .
The automaton reads words from left to right. A finite path in the automaton, , is word in , the set of transitions: such that for and ; however, we usually denote the paths using only the labels, that is, . We say that the path terminates at the state and we say that the path passes through the state, if there is such that , in (1). See [12] for more details on automata theory.
Theorem 1 (see [11, 13]). Let be a nonempty prefix of , the fixed point of the substitution . Then there exists a unique path on the prefix automaton of : such that and . Conversely, to any such path, there corresponds a prefix of , given by the above formula.
Due to this theorem, we can associate numeration systems for the positive integers, in the following way. Let be a positive integer and the prefix of length of ; we define , by , where and . Moreover, this representation is unique. This numeration system is called the Dumont-Thomas numeration system associated with the substitution (cf. [13, 14]). For general theory of numeration systems see [8, 14–18].
3. Semigroups
We consider the following subset of positive integers:
We define “”, a binary operation in , in the following way: let such that and , so This binary operation is well defined since the paths in the prefix-automaton, associated with the elements of , terminate at the initial state, so we can concatenate them. It is clear that the operation is associative. Hence, is a semigroup.
However, the binary operation is not always commutative; for example, in the Fibonacci case, we consider and , so and , so and .
If the word starts with the symbol , then in the prefix automation the transition is allowed, so is a path in , and By convention we say that and let . Clearly , but ; since , it does not have to be . Hence, the pair has a left identity. If and , we always consider that . However is not a semigroup; consider ; since and on the other hand , therefore . The algebraic structure of the pair is an unital, a set with a closed binary operation with identity; see [19].
We shall change the definition of the binary operation so that we have a semigroup with identity. Without loss of generality, we can suppose that the substitution is of the form , where is a finite word on the alphabet and or ; that is, is the empty word or a word consisting only of a concatenation of the symbol . Let be the set of natural numbers whose representation is , where the is a path in that starts in the initial state and terminates in the state and the path terminates at the initial state.
We can think of the set as the set of numbers whose representation is a path in the prefix-automaton that starts in the state and terminates in the state , such that it passes through in the step before terminating.
We define a binary operation “” on as follows. Let , whose representations are and ; so This binary operation is well defined since it is an element of . The operation is associative. Let such that , and ; so
The semigroup has a left identity; however, there might not be more than one left identity element. Let be any transition in the prefix automaton from the state to the state , and such that . By definition and , for any . In general does not have a right identity.
Proposition 2. Let be a primitive substitution on such that where or and is a finite word on the alphabet. If the symbol does not belong to the word , then the identity of the semigroup is the element of whose representation is , with if , or , if .
Proof. Let if , or , if . We remark that is a proper prefix of , so , and in the prefix automaton; that is, is a transition from the state to the state . Since does not have the symbol , is the only transition in the prefix automaton from the state to . The transition in the automaton from the initial state to itself and from the state to the state are shown in Figure 2; in this figure we suppose that the word does not contain the symbol .
Let with . We have . Since , is a transition from the state to and there is only one transition from state to state , so . Hence, . On the other hand, is a left identity, as it was pointed out before.
We remark that if and only if is the empty word. For example, this is the situation in the -bonacci substitutions.
We say that a semigroup is finitely generated if there exists with such that for all we have , for some and , . The elements of are called generators of .
We remark that in the Fibonacci case, the set is not finitely generated because we have the paths that correspond to elements in but the word is not in . Similarly the semigroup is not finitely generated; considering the paths , with , the numbers associated with these paths cannot be finitely generated since is not in . The following theorem shows that this is a general situation.
Theorem 3. The semigroups and are not finitely generated.
Proof. We suppose, as before, that , where or , with , and is a finite word in the alphabet. Let if or if . It is a proper prefix of and is a transition in the prefix automaton. Let be a path in the prefix automaton such that it starts and ends at the state so that does not terminate at the state , for . Let be the number in , such that , with . We consider the numbers , with ; by definition they belong to . These numbers cannot be finitely generated, in fact. Let us consider ; if we want to write it as , then has to be of the form , but this is not a representation given by the Dumont-Thomas numeration system, since the first symbol is . There is no other possible “factorization” of , since for . Therefore, cannot be written as product of , different from itself, and the identities. Similar argument holds for for . Therefore, is not finitely generated.
The proof that is not finitely generated follows in a similar manner.
We generalize the classical concepts of prime and composite numbers into this context. We say that an element of is -composite if there exist different of an identity element such that . We say that is -prime if it is not composite. In the proof of Theorem 3, we have shown that the numbers are -primes.
Corollary 4. The number is -prime if and only if where is a path in the automaton with the property that does not terminate at the state for .
Proof. Let whose Dumont-Thomas representation is where is a path in such that does not terminate at the state for . So the numbers given by are not in . Since the representation is unique, there are not such that . Hence is -prime.
On the other hand, let be -prime such that . If there is a such that is a path in that terminates at the state , then , where , , and is a transition in the automaton from the state to the state . By construction and are in ; therefore, could not be prime.
We point out that the semigroup acts on the set of the positive integers. Let , and ; we suppose that . If and , where is a path in the prefix automaton, , that terminates at the state , then is a path in , which starts at the initial state and terminates at the state . So is defined and ; however, is not in , if . Let be the map defined by . This map defines a left action of on , since is the identity, where is a left identity of , and .
The dynamics of this action can be studied. Let ; the orbit of under is the set , which is denoted by . Similarly we define the orbit of under . The following proposition shows how the orbits of both actions are related.
Proposition 5. Let be a positive integer. Then .
Proof. Let be the representation of , where is a path in the automaton that starts in the initial state and terminates at the state . Let be an element of , so , for some , with , and ; so . Since is a path in the automaton that starts and terminates at the initial state, we have , where . If , then , where , with being a transition between the states and . Hence .
Let , since does not have identity . So , for some . Let where is a path in the automaton that starts and terminates at the initial state. Hence, . Let be an element of , whose representation is , where is a transition from the state to . So . Therefore, . Clearly , since , where .
A further step in the study of the dynamics of this action is the characterization of its invariant sets. The set is -invariant if .
4. Density of the Semigroups
In this section, we will study the density of the sets and in the set of positive integers. Let be the Perron-Frobenius eigenvalue of and and the right and left nonnegative eigenvectors associated with ; that is, , . If we choose them so that , then the matrix is called the Perron projection of (cf. [20, 21]).
Proposition 6. An estimation of the density of in the set of positive integers is given by where is the left positive eigenvector, associated with Perron-Frobenius eigenvalue, of the transition matrix .
Proof. Let be the entry of the matrix . The number of paths of length in that start in the state and end in the state is given by .
Let be the number of paths of length in the automaton that start at the initial state; that is, . By the Perron-Frobenius Theorem for primitive matrices (cf. [21]). So the asymptotical growth of is given by .
Let be the number of paths of length in , that start in the state and terminate in the state , so . So its asymptotical growth is given by . Therefore
This expression simplifies in the following way:
We prove in a similar manner the corresponding result for .
Corollary 7. An estimation of the density of in the set of positive integers is given by where is as in Proposition 6.
Example 8. In the tribonacci case, the transition matrix is
The Perron-Frobenius eigenvalue is , the real root of the polynomial . The left eigenvector of is , where and the right eigenvector of is , without normalization. According to Proposition 6 and Corollary 7, the upper bound of the density of is and that of is .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.