1. Introduction and Preliminaries
In the classical theory of theoretical computer science, an automaton is a finite state machine which accepts certain strings of letters from a fixed base alphabet. Such strings are usually called
words, and sets of words are called
languages. Formal language theory is the study of properties of languages and automata. One of a generalized study of formal languages is the study of tree languages, i.e., the words of the classical case can be considered as a particular kind of terms. Since terms are commonly represented by tree diagrams, such formal languages are also called
tree languages. For more details in this background, see [
1,
2,
3]. We now recall the fundamental notion of terms as follows: Let
, for
n a natural number, be a finite set which elements are called
variables and
be countably infinite. The variable
in
is an alphabet of formal languages. To define terms from this alphabet, we use a set
of operation symbols, indexed by the set
I. The type is the sequence
of the natural numbers which are arities of the operation symbols
. An
n-ary term of type τ is defined inductively by: Every variable
is an
n-ary term of type
and
is an
n-ary term of type
where
are
n-ary terms of type
and
is an
-ary operation symbol. The set of all
n-ary terms of type
is denoted by
. Let
be the set of all terms of type
. See [
4,
5,
6,
7,
8,
9,
10] for other related topics of terms.
Every term can be represented by a tree diagram, i.e., we consider term as a rooted tree whose vertices correspond to the operation symbols and variables that occur in it. We call this tree the
tree representation of a term t. For instance, a tree representation of a term
can be shown in the below diagram.
Substituting the variables that appear in a term by the other terms, one obtains a new term. This can be described by the
-generalized superpostion
,
defined inductively by the following steps: for
- (1)
If ; , then
- (2)
If ; , then .
- (3)
If , then
Then, we can form the algebraic structure,
-ary algebra
consisting the universe
together with one
-ary operation
. In [
11], the algebra in this form is known as a Menger algebra with infinitely many nullary operations. It is not hard to show that this algebra satisfies the superassociative law:
for all arbitrary terms
Using the concepts of terms and generalized superposition, the following ideas are essentially recalled. It was known from [
12] that a formal definition of a strong hyperidentity and a strong solid variety can be given using the concept of a generalized hypersubstitution. We now recall such concept as follows: Let
be an indexed set of operation symbols of type
where
is
-ary,
is a natural number. Let
be the set of all arbitrary mappings
which is called a
generalized hypersubstitution of type τ. To define a binary operation on this set, the essential defining is necessary. Any
can be uniquely extended to a mapping
which is defined by
- (1)
- (2)
where , are already defined.
Using the extension of generalized hypersubstitution of type
, the binary operation
is defined by
where ∘ denotes the usual composition of mappings. The generalized hypersubstitution
, which sends each
to the term
, is an identity element for
. Then,
is a monoid. See the following references for the research topics and current trends in this direction [
13,
14,
15,
16,
17,
18].
Let
be the power set of all the set of all terms of type
. An inductive definition of an
-ary operation on
is completely defined in [
19]. Let
n be a natural number,
are arbitrary subsets of
. Then, an
-ary generalized superposition operation
is defined inductively by
- (1)
If ; , then .
- (2)
If ; , then .
- (3)
If , and suppose that for all are already defined, then .
- (4)
If
B is an arbitrary nonsingleton subset of
, then
If one of the sets
is empty, we define
. It turns out that the following algebra of type
,
is a Menger algebra and called a power Menger algebra with infinitely many nullary operations. Then, it was proved that this algebra satisfies the superassociative law [
19].
A non-deterministic generalized hypersubstitution of type
[
19] is a mapping
. The set of all such mappings is denoted by
. It is well-known that every
generates a mapping
which takes a tree language into itself by the following inductive way:
- (1)
,
- (2)
where is a variable from X,
- (3)
if are already defined,
- (4)
if B is an arbitrary nonsingleton subset of
The algebraic properties of
were proved in [
19] that every extension
of non-deterministic generalized hypersubstitution
is an endomorphism of the algebra
. Applying an extension
, in [
20], the binary operation
was introduced by setting
. Moreover, the non-deterministic generalized hypersubstitution
was defined to be an identity element where
for all
. As a consequence, the structure
forms a monoid.
As we mentioned above, there are many authors who studied the concepts of terms and tree languages in various structures. The lack of relationships among the algebras of terms and tree languages allows us to study in this paper. Thus, two potential questions natually arise: Firstly, are there interactions between several algebras of terms and tree languages? Secondly, how can these connections be described in terms of embedding, if any? In the present paper, these problems are answered in
Section 2. Particularly, in
Section 3, we define another binary operation for the set of all non-deterministic generalized hypersubstitutions and provide their interesting properties. An embedding theorem of a Menger algebra with infinitely many nullary operations in a suitable algebra is studied in
Section 4. We also give a concrete example of this representation.
2. Monomorphisms between Semigroups of Terms and Tree Languages
In any Menger algebra
, i.e., the structure consisting of a nonempty set with an
-operation satisfying the superassociative law, a binary operation + on
G can be defined by
. It is not difficult to see that + is associative. The algebra
is called the
diagonal semigroup. Moreover, on the Cartesian power
, one can define a binary operation ∗ by
It turns out that
forms a semigroup, which is called the
binary comitant. In 2006, Denecke and Jampachon [
11] defined these two semigroups for the set of terms and characterized the regularity and Green’s relations. The situation for tree languages was studied in [
21].
In order to study the connection between the semigroups of terms and tree languages, we need the following lemma:
Lemma 1. For any terms , we have Proof. We give a proof by induction on the complexity of the term
t. The proof is easy to verify if
t is a variable. Assume that
and
for every
. Then,
□
Let I be a nonempty indexed set. The symbol means a fixed n-ary type with operation symbols . That is, is a sequence of -tuple of fixed n-ary operation symbols. For instance, if , . This means that we have three binary operation symbols.
On the set , the binary operation can be defined by for all . Similarly, the binary operation on is defined by setting for all Then, we have
Proposition 1. The diagonal semigroup can be embedded into the diagonal power semigroup .
Proof. It follows immediately from the superassociativity of and that and are semigroups. For any term t in , we define a mapping by . Obviously, is injective. Moreover, the homomorphism property is also true by applying Lemma 1. Since, for all , we have □
Proposition 2. The binary comitant semigroup where on is defined bycan be embedded into the semigroup where the operation on is defined by Proof. Clearly, and are semigroups since the operations and satisfy the superassociative law, respectively. A mapping can be defined by for all terms of type . It is clearly that is injective. To prove that is homomorphism, we let be elements in . Then, by Lemma 1, we obtain
This finishes the proof. □
A relationship between the diagonal semigroup and the binary comitant semigroup is now provided.
Proposition 3. Let be a mapping which is defined by for all . Then, φ is a monomorphism from into , i.e., it satisfies the identity for all .
Proof. Injectivity of is clearly obtained. In addition, is homomorphism since . □
Using the generalized superposition
for every
, we define a binary operation
on
in the following way:
for all
. Then, we have
Proposition 4. The set with a binary operation forms a monoid.
Proof. Because the generalized superposition satisfies the superassociative law, then a binary operation is associative. We see that a variable acts as the identity element, by the property of the generalized superposition , we have . □
The situation for tree languages was already defined in [
22], i.e., for any subsets
of
, the binary operation
on
is defined by
Furthermore,
is a monoid, since the singleton
is an identity with respect to
.
Proposition 5. The monoid can be embedded into .
Proof. Firstly, we define a mapping by for all . Clearly, is injective. It can be shown straightforwardly that a mapping is a homomorphism. □
3. The Left Seminearring of Non-Deterministic Generalized Hypersubstitutions
In this section, we investigate the structural properties of non-deterministic generalized hypersubstitutions. One of the algebraic structures that we are interested in is a seminearring.
Definition 1. A left (right) seminearring is a triple of a nonempty set R together with two binary operations, denoted by + and ·, respectively, such that + and · is associative on R, and satisfying the left (right) distributive law, i.e., for all
It was proved in [
23] that, under suitable two binary operations, i.e.,
and
, which are defined by
for all
; the set of all generalized hypersubstitutions of type
forms a left seminearring.
On the diagonal semigroup
as we mentioned in the previous section, we now establish the relationship between the diagonal semigroup and the semigroup
. To do this, we improve a definition of a binary operation
by the following
It can be proved by a direct calculation and the superassociativity of the generalized superposition that this defining operation is associative. Then, we have that
Proposition 6. is a semigroup.
A strong connection between the diagonal semigroup and the semigroup of all generalized hypersubstitutions is provided in the following theorem:
Theorem 1. The diagonal semigroup can be embedded into .
Proof. For any generalized hypersubstitution of type and any term t, the mapping is defined by , where is an n-ary operation symbol. To prove this theorem, we define a mapping which takes each n-ary term t to a generalized hypersubstitution of type by for all . It is obvious that is injective. Now, let . Then, In fact, we have □
The next purpose of this section is to define another binary operation on the set of all non-deterministic generalized hypersubstitutions. This leads us to form a novel semigroup. Now, we introduce a novel definition of the binary operation
on the set of all non-deterministic generalized hypersubstitutions of type
by setting
Then, forms a semigroup with respect to the binary operation , which is presented in the following proposition:
Proposition 7. is a semigroup.
Proof. Let
and
be elements in
. Because of the superassociativity of the generalized superposition
, we have
| = | |
| = | |
| = | |
| = | |
The proof is finished. □
Then, we have
Theorem 2. is a left seminearring.
Proof. It is well known that is a semigroup and by Proposition 7 is also a semigroup. Now, we show the left distributivity. For this, let and be non-deterministic generalized hypersubstitutions of type . Then, by the fact that any extension of non-deterministic generalized hypersubstitutions of type is endomorphism on the set of tree languages, we have
| = | |
| = | |
| = | |
| = | |
| = | |
This shows that the left distributive law is valid. □
The following counterexample shows that the right distributive law is not satisfied.
Example 1. Let I be a singleton and the type with one binary operation symbol f. Define non-deterministic generalized hypersubstitutions of type , say and by Consider | = | |
| = | |
| = | |
| = | |
and
Thus, This means that the right distributive law is not true.
Based on the monoid
, various algebraic structural properties of some submonoids were investigated by Leeratanavalee in [
24]. Now, we extend these concepts to non-deterministic generalized hypersubstitutions and study some interesting relationships.
Definition 2. Let be a type with operation symbols having the arity for each . A non-deterministic generalized hypersubstitution σ is said to be
- (1)
projection non-deterministic generalized hypersubstitution if the image is a nonempty subset of X. Let be the set of all projection non-deterministic generalized hypersubstitutions of type τ.
- (2)
a pre-non-deterministic generalized hypersubstitution if is a nonempty subset of . Let be the set of all pre-non-deterministic generalized hypersubstitutions of type τ.
Theorem 3. and are submonoids of .
Proof. Obviously, the identity non-deterministic generalized hypersubstitution belongs to the sets and . Now, we let . To prove that , we consider the following four cases:
Case 1: and . Then, where . Consider . By the assumption, we have Thus,
Case 2: and . Then, where . Thus, .
Case 3: . Obvious.
Case 4: . Then, both and are nonempty subsets of X. Thus, for some and .
Finally, it is not difficult to verify that the composition of two pre-non-deterministic generalized hypersubstitutions is again a pre-non-deterministic generalized hypersubstitution: □
Theorem 4. and form sub-left seminearrings of .
Proof. Our goal is to show that the sets and are closed under the binary operation . Let . Then, the images of them are nonempty subsets of X. Since and are nonempty subsets of X, is also a set of some variables from X. Thus, . Next, let . Then, the images of them are nonempty subsets of . Since and are nonempty subsets of , is a nonempty subset of . This shows that is a pre-non-deterministic generalized hypersubstitution. □
Similar to Theorem 1, the situation for tree languages is given. To do this, the following tool is needed. For every subsets
of terms of type
, a binary operation
is defined by
Using the fact that the generalized superposition over tree languages satisfies the superassociative law, then we get
Lemma 2. forms a semigroup, and called the diagonal power semigroup.
We now investigate the relationship between the diagonal power semigroup which is constructed in Proposition 2 and the semigroup .
Theorem 5. The diagonal power semigroup can be embedded into , .
Proof. Let be a non-deterministic generalized hypersubstitution of type and B be a subset of . Then, we associate a mapping by , where is an n-ary operation symbol. Then, a mapping from the powerset of n-ary terms to the non-deterministic generalized hypersubstitution of type is defined by for all . It is commonly seen that is injective. Now, let . Then, In fact, Now, the proof is completed. □
We complete this section by giving a significant connection between generalized hypersubstitutions and non-deterministic generalized hypersubstitutions. For convenient, a non-deterministic generalized hypersubstitution will be denoted by .
Lemma 3. For any term t, a generalized hypersubstitution of type τ σ, and a non-deterministic generalized hypersubstitution , Proof. We give a proof on the complexity of the term t. If t is a variable, we are done. The proof follows directly from Lemma 1 if □
Theorem 6. can be embedded into
Proof. Let
be an
-ary operation symbol and
a generalized hypersubstitution of type
. Then, we define
by
. It is clear that
belongs to
. In order to prove that there is a monomorphism from
to
, we define a mapping
by
for all
Clearly, the mapping
is injective. Next, we let
be two elements in
. Then, by Lemma 3, we have
This shows that
is a homomorphism. □
Corollary 1. The left seminearring can be embedded into the left seminearring .
A significant consequence of Theorem 6 is that a mapping in corresponds to the idempotent or a regular non-deterministic generalized hypersubstitution in
4. Representation of Menger Algebra with Infinitely Many Nullary Operations by n-Ary Functions
Firstly, we recall some preliminaries and background notions of
n-ary functions, see [
25,
26,
27]. Let
be the
n-th Cartesian product of a nonempty set
A. Any mapping from
to
A is called a
full n-ary function or an
n-ary operation. The set of all such mapping is denoted by
. One can consider the
Menger’s composition on the set
, i.e., an
-operation
defined by
where
The set
is said to be an
algebra of full functions or
algebra of operations if the composition of
functions from this set is also in this set, i.e., closed with respect to Menger’s composition. We can remark here that the Menger’s composition can be reduced to the usual composition of functions if
.
A Menger algebra of all full n-ary functions or Menger algebra of all n-ary operations is a pair of the set of all full n-ary functions defined on A and the Menger composition of full n-ary functions satisfying the superassociative law. Each subalgebra of this algebra will be called a Menger algebra of full n-ary functions or Menger algebra of n-ary operations.
For each term
t of the algebra
, we associate the full
n-ary function
by putting
for all
, where
is an
-ary generalized superposition operation defined on
. The full
n-ary function
is an element of
, which is called an
inner left translation of
corresponding to the term
t of
.
In order to prove our main theorem, the following lemmas are primarily essential.
Lemma 4. Let be a Menger algebra. Then,for all , where and are generalized superposition operation and Menger’s composition, respectively. Proof. Let
be arbitrary elements in
. Then, we have
| = | |
| = | |
| = | |
| = | |
| = | |
The proof is completed. □
Now, we let . Our next aim is to show that forms a Menger subalgebra of .
Lemma 5. The set forms a subalgebra of and thus is a Menger algebra of full n-ary functions.
Proof. Obviously, . Let be arbitrary full n-ary functions in . It follows immediately from Lemma 4 that the composition of such mappings again a full n-ary function. □
We now establish the Cayley’s theorem of a Menger algebra with infinitely many nullary operations as follows:
Theorem 7. Let be a Menger algebra with infinitely many nullary operations. Define a mapping byfor all . Then, ψ is an isomorphism from to and so . Proof. Clearly, is surjective. By Lemma 4, we have and thus is a homomorphism. Furthermore, is injective. Indeed, suppose that . Then, we obtain . By the definition of generalized superposition , we conclude that t and s are the same term. Hence, is injective. Therefore, is an isomorphism from to . □
Finally, an interesting concrete example is demonstrated by considering an indexed set I is singleton and setting the type with a binary operation symbol f and a natural number n equal to 1. This means that we have the algebra where . Moreover, the full n-ary function also reduces to the usual full transformation with one composition of functions.
Example 2. Consider a semigroup with respect to a binary operation which is defined by the following table: | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
It is observed that A is a semigroup containing an identity element . We now illustrate that A is isomorphic to some sets of the full transformation semigroups. To do this, we first demonstrate the process to establish a full transformation by considering a mapping such that , , and . Thus, . Furthermore, a mapping is defined by , , and . Thus, the full transformation which corresponding to a term in A isand so . For other terms in A, we obtain By Theorem 7, . Furthermore, the table for these representations is just like the original table with a term t renamed by , as seen in the following table: | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
We can comment here that, if we put , then this set is also an infinite subsemigroup of and then every term in this set is a left zero and idempotent. Hence, forms a left zero band. We also have that it is a right ideal of and thus is not a right simple semigroup.