Abstract
In this paper we consider two classes of posets labeled over an alphabet A. The class 
is built from the letters and closed under the operations of series finite, 
and 
products, and finite parallel product. In the class 
, 
and 
products are replaced by 
and 
powers. We prove that 
and 
are freely generated in their respective natural varieties of algebras 
and 
, and that the equational theory of 
is decidable.
Keywords: Transfinite N-free posets, Series-parallel posets, Variety, Free algebra, Series product, Parallel product, 
-power, 
-power, Decidability
Introduction
In his generalization of the algebraic approach of recognizable languages from finite words to 
-words, Wilke [22] introduced right binoids, that are two-sorted algebras equipped with a binary product and an 
-power. The operations are linked together by equalities reflecting their properties. These equalities define a variety of algebras. This algebraic study of 
-words have since been extended to more general structures, such as for example partial words (or equivalently, labeled posets) or transfinite strings (long words). In [8], shuffle binoids are right binoids equipped with a shuffle operation that enables to take into consideration N-free posets with finite antichains and 
-chains instead of 
-words. In [3, 5], the structure of right binoids in two parts is modified in order to enable products to extend over 
, ie. small ordinals (
, 
) and countable ordinals. The latter algebras are enriched in [10, 11] with operations such as for example reverse 
-power in order to take into account countable linear orderings (scattered in some cases). Some of the previous algebraic enrichments were also applied to shuffle binoids [4, 12]. The motivations in [3–5, 10, 11, 17, 22] are mainly the study of the links between automata, rational expressions, algebraic recognition and monadic second-order logic. In [7–9, 12, 22] the authors focus essentially on varieties of algebras; for example, free algebras are characterized in the corresponding varieties, and decisions algorithms for equivalence of terms are provided.
Let us denote by 
the reverse ordering of 
. In this paper we focus on algebras equipped with a parallel product, series product, and either 
and 
products or 
and 
powers. For example, the class 
of N-free posets in which antichains are finite and chains are scattered and countable orderings lies in this framework. In [2, 6] this class has been studied from the point of view of automata, rational expression and logic. We prove here that 
is the free algebra in a variety 
of algebras equipped with a parallel product, series product, and 
and 
products. By removing the parallel product, it follows that 
, the class of scattered and countable words over A, is also a free algebra in the corresponding variety. We also consider the class 
where the 
and 
products are replaced by 
and 
powers, and show that it is freely generated in the corresponding variety 
. Relying of decision results of [2] we prove that the equality of terms of 
is decidable.
Linear Orderings and Posets
We let 
denote the cardinality of a set E, and [n] the set 
, for any non-negative integer 
.
Let J be a set equipped with a strict order <. The ordering J is linear if either 
or 
for any distinct 
. We denote by 
the backward linear ordering obtained from the set J with the reverse ordering. A linear ordering J is dense if for any 
such that 
, there exists an element i of J such that 
. It is scattered if it contains no infinite dense sub-ordering. The ordering 
of natural integers is scattered as well as the ordering 
of all integers (negative, 0 and positive). Ordinals are also scattered orderings. We let 
, 
and 
denote respectively the class of finite linear orderings, the class of countable ordinals and the class of countable scattered linear orderings. We also let 0 and 1 denote respectively the empty and the singleton linear ordering. We refer to [20] for more details on linear orderings and ordinals.
A poset
is a set P partially ordered by <. For short we often denote the poset 
by P. The width of P is 
where 
denotes the least upper bound of the set. In this paper, we restrict to posets with finite antichains and countable and scattered chains.
Let 
and 
be two disjoint posets. The union (or parallel composition) 
of 
and 
is the poset 
. The sum (or sequential composition) 
is the poset 
. The sum of two posets can be generalized to a J-sum of any linearly ordered sequence 
of pairwise disjoint posets by 
. The sequence 
is called a J-factorization, or (sequential) factorization for short, of the poset 
. A poset P is sequential if it admits a J-factorization where J contains at least two elements 
with 
, or P is a singleton. It is parallel when 
for some 
. A poset is sequentially irreducible (resp. parallelly irreducible) when P is either a singleton or a parallel poset (resp. a singleton or a sequential poset). A sequential factorization 
of 
is irreducible when all the 
are sequentially irreducible. It is non-trivial if all the 
are non-empty. The notions of irreducible and non-trivial parallel factorization are defined similarly. A poset is scattered if all its chains are scattered. The class 
of series-parallel scattered and countable posets is the smallest class of posets containing 0, the singleton, and closed under finite parallel composition and sum indexed by countable scattered linear orderings. By extension of a well-known result on finite posets [18, 21], it has a nice characterization in terms of graph properties: 
coincides with the class of scattered countable N-free posets without infinite antichains [6]. Recall that 
is N-free if there is no 
such that 
.
When 
and 
or 
for some 
then 
is a factor of P; the factors of 
and S are also factors of P.
F. Hausdorff proposed in [16] an inductive definition of scattered linear orderings. In fact, each countable and scattered linear ordering is obtained using sums indexed by finite linear orderings, 
and 
. This has been adapted in [6] to 
.
We let 
denote the closure of a set E of posets under finite disjoint union and finite disjoint sum.
Definition 1
The classes of countable and scattered posets (equivalent up to isomorphism) 
and 
are defined inductively as follows:
 |
and the class 
of countable and scattered posets by 
.
The following theorem extends a result of Hausdorff on linear orderings [16].
Theorem 1
([6]).
.
For every 
, 
can be decomposed as the closure of 
by finite disjoint union and finite disjoint sum:
Theorem 2
([6]). For all 
, 
, let
 |
Then 
.
Example 1
is the set of all finite N-free posets. Its subset 
is the set of all finite linear orderings. The linear orderings 
and 
are contained in 
. Each poset of 
has some chain isomorphic to either 
or 
, but can not have a chain isomorphic to 
and another isomorphic to 
. The ordering 
of all integers is in 
. For all 
, 
.
Define a well-ordering on 
by 
if and only if 
or 
and 
. As a consequence of Theorems 1 and 2, for any 
there exists a unique pair 
as small as possible such that 
.
Definition 2
The rank
r(P) of 
is the smallest pair 
such that 
.
Example 2
The linear ordering 
has rank 
. Each linear ordering I of 
has rank 
for some 
. For all 
, 
.
Remark 1
Let 
with 
, 
. Assume that 
is a non-trivial J-factorization of P for some 
. If 
(resp. 
), then, for all 
, 
. In addition, for all 
such that 
, for all 
there exists 
such that 
(resp. 
) and
 |
This implies that, for all 
, 
(resp. 
) is of rank 
.
Lemma 1
Let 
be a sequential poset such that 
, 
. Let 
and 
be some non-trivial J- and 
-factorizations of P where 
. Then 
.
Proof
Assume by contradiction that 
. Assume wlog that 
and 
. Let 
and 
for some 
. Then 
. As a consequence of Remark 1, 
. Observe that there exists 
such that R is a sequential factor of 
. Let 
. As a consequence of Remark 1, 
. Furthermore, 
. Thus 
too. We have 
, and by Theorem 2, 
, which is a contradiction.
In [6] an equivalence relation 
over the elements of a poset of 
is given, such that 
is isomorphic to a countable and scattered linear ordering (Lemma 9), and such that each equivalence class is a sequentially irreducible factor of P (Lemma 10). This leads to the following proposition.
Proposition 1
([6]). Each poset of 
admits a unique irreducible sequential factorization.
Definition 1 and Theorem 1 provide a well-founded definition of 
which we consider from now as a set, although originally defined as a class.
Labeled Posets
An alphabet A is a non-empty set (not necessarily finite) whose elements are called letters or labels. In the literature a word over A is a totally ordered sequence of elements of A. The sequence may have properties depending on the context, for example it can be finite, an ordinal, or a countable scattered linear ordering. The notion of a finite word has early been extended to partial orderings (finite partial words or pomsets [14, 15, 23]). In this paper we consider a mixture between the notions of finite partial words and words indexed by scattered and countable linear orderings.
A poset P is labeled by A when it is equipped with a labeling total map 
. Also, the finite labeled posets of width at most 1 correspond to the usual notion of words. We let 
denote the empty labeled poset. For short, the singleton poset labeled by 
is denoted by a, and we often make no distinction between a poset and a labeled poset, except for operations.
The sequential product (or concatenation, denoted by 
or 
for short) and the parallel product
of two labeled posets are respectively obtained by the sequential and parallel compositions of the corresponding (unlabeled) posets. By extension, the sequential product 
of a linearly ordered sequence of labeled posets is the poset 
in which the label of the elements is kept. In particular, the 
-product (resp. 
-product) of an 
-sequence (resp. 
-sequence) of labeled posets 
(resp. 
) is denoted by 
(resp. 
). The 
-power (resp. 
-power) 
(resp. 
) of the poset P is the 
-product (resp. 
-product) of an 
-sequence (resp. 
-sequence) of posets that are all isomorphic to P. As usual, in this paper we consider two labeled posets to be identical if they are isomorphic. By extension, the rank r(P) of a labeled poset P is the rank of its underlying unlabeled poset.
Let A and B be two alphabets and let P be a poset labeled by A. For all 
, let 
be some poset labeled by B, and let 
. The poset labeled by B consisting of P in which each element labeled by the letter a is replaced by 
, for all 
, is denoted by 
. If the underlying posets of P and of all the 
are in 
, then so is 
.
Definition 3
Let A be an alphabet. We define:
, the smallest set of posets labeled by A containing 
, a for all 
, and closed under operations of sequential, parallel, 
and 
-products. According to Theorem 1, the underlying posets are precisely those of 
;
, the smallest subset of 
containing 
, a for all 
, and closed under operations of sequential product, 
-power and 
-power;
, the smallest subset of 
containing 
, a for all 
, and closed under operations of sequential product, 
-product and 
-product;
, the smallest subset of 
containing 
, a for all 
, and closed under operations of sequential and parallel product, 
-power and 
-power;
, the smallest subset of 
containing 
, a for all 
, and closed under operations of sequential product, parallel product and 
-product (note that there is no 
-product here).
Note that 
and 
.
Varieties
In this section we define the different varieties studied throughout this paper by listing the axioms they satisfy. The usual notions and results of universal algebra apply to our case, even if we use here for example operations of infinite arity. For more details about universal algebra, we refer the reader to [1]. In the following 1 is considered as a neutral element (the interpretation of a constant).
for all 
-sequences 
and all decompositions
for all 
-sequences 
and all decompositions
Definition 4
We define
, the collection of algebras 
satisfying the axioms ()–(), ()–() and ()–();
, the collection of algebras 
satisfying the axioms (), (), () and ()–();
, the collection of algebras 
satisfying the axioms ()–(), ()–() and ()–();
, the collection of algebras 
satisfying the axioms ()–();
, the collection of algebras 
satisfying the axioms (), ()–() and (),().
In order to simplify the notation, an algebra whose set of elements is S is sometimes denoted by S when there is no ambiguity.
Freeness
Throughout this section, A denotes an alphabet. We start by proving the freeness of 
.
Theorem 3
is freely generated by A in 
.
Proof
For all 
, let 
denote the set of posets of 
of rank 
or less. Let 
be any algebra of 
and let 
be any function. We show that h can be extended into a homomorphism of 
-algebras 
in a unique way. Define 
as 
where each 
is defined by induction over 
as follows. Let us denote by 
. Let 
. If 
then 
. Otherwise
if
and 
then 
;- if
:- if P is a sequential poset then it has a factorization
where each
20 
is a non-empty poset of rank lower than 
and 
. Define 
by 
- otherwise, P is a parallel poset. Write
where each
is a sequential poset and 
. Then, define 
by 
By Theorem 2, the factorizations used in the definition of 
exist. However, observe that the sequential ones ((19) and (20)) are not unique. This would question the fact that 
is a well-defined function. For all 
of rank 
, we show that:
does not depend on the factorization of P and thus is well-defined;
commutes with all the operations of 
: 
, for some 
;
, for some 
.
We proceed by induction on 
. Let us start by proving that 
maps 
to the same element of M regardless of the factorization of P. If 
the theorem follows immediately. Otherwise, assume first that 
. By Lemma 1, all the possible factorizations of P as in (19) are either all 
-factorizations or all 
-factorizations. Assume wlog that P admits only 
-factorizations as in (19). Let 
and 
be two different such 
-factorizations. By definition of 
 |
There exists a sequence 
of non-empty posets such that 
and for all 
there exist 
such that
 |
By induction hypothesis 
commutes with all the operations of 
. Then, we have for all 
:
 |
Thus 
can be written as
 |
We have 
. The case where P admits only 
-factorizations as in (19) is proved symmetrically using () instead of (). In addition, using () instead of () and arguments similar to those of the previous case, we prove that when P is sequential and 
, 
does not depend on the factorization of P.
Thus, we have proved that 
is well-defined for sequential posets of rank 
. In addition, the irreducible parallel factorization is unique modulo the commutativity of 
. Thus 
is well-defined for all posets of rank 
, for all 
. Furthermore, proving that 
commutes with all the operations in 
can be done by induction on r(P) too. The arguments are very similar to those used to prove that 
is well-defined. It follows that 
is a homomorphism of 
-algebras. In addition, since 
relies on h then 
is unique.
The proofs of the following theorems rely on the same arguments. It suffices to restrict 
to the operations of the corresponding variety. In particular, this provides a new proof of Theorem 5.
Theorem 4
is freely generated by A in 
.
Theorem 5
([12]).
is freely generated by A in 
.
In the remainder of this section, we prove the freeness of 
in 
. The arguments are similar to those of the proof of Theorem 6.1 in [12] in which the variety considered is 
without 
-power. We need the following result.
Theorem 6
([9]).
is freely generated by A in 
.
Lemma 2
Let A and B be two alphabets. Let 
such that S is closed under sequential product, 
-power and 
-power. Let 
be some function defined by 
for some 
. Then, the function 
extending f defined by 
, for all 
, is a homomorphism from 
to 
.
Furthermore, if 
is bijective, S is generated by G, and G contains only sequentially irreducible posets then 
is bijective.
Proof
Let 
whose irreducible sequential factorization is 
for some 
, where each 
. Note that
 |
Let 
and 
be some sequential factorizations of u. Then, one can prove easily that
 |
relying on the uniqueness of the irreducible sequential factorization of u (Proposition 1).
Let us prove now that when f is bijective and S is generated by a set of sequentially irreducible posets then 
is bijective. Let 
and assume that 
and 
. Let 
and 
be the irreducible sequential factorizations of respectively u and v, for some 
, where each 
and 
are in A. By definition of 
, 
and 
where each 
and 
. Then, for all 
and for all 
, 
and 
are sequentially irreducible posets of G. Assume that 
. Then 
and, for all 
, 
. We have, for all 
, 
since 
is injective by hypothesis. In addition, as G generates S, each element P of S can be written as 
where each 
, for some 
. Since 
is surjective by hypothesis, for all 
there exists 
such that 
. Then 
.
As a consequence of HSP Birkhoff’s Theorem (see eg. [1, Theorem 1.3.8]) and Lemma 2:
Corollary 1
For all 
closed under sequential product, 
-power and 
-power and generated by a set of sequentially irreducible posets of 
, 
is a 
-algebra.
Corollary 2
For all 
closed under sequential product, 
-power and 
-power and generated by a set G of sequentially irreducible posets of 
, 
is freely generated by G in 
.
We are now ready to prove the following theorem.
Theorem 7
is freely generated by A in 
.
Proof
For all 
, let 
be the subset of 
consisting all its posets of width lower or equal to i. Then 
. Note that 
and 
. Observe that for all 
, 
is closed under sequential product, 
-power and 
-power. In addition, for all 
, 
is generated by its sequentially irreducible posets. By Corollary 1, for all 
, 
can be considered as a 
-algebra. In addition, by Corollary 2, for all 
, 
is freely generated by its sequentially irreducible posets in 
. Then, for all 
and 
, a function 
can be extended in a unique homomorphism of 
-algebras 
.
Let S be some 
-algebra and let 
be some function. We show that h can be extended into a homomorphism of 
-algebras 
in a unique way. Indeed, we define 
as 
where each 
is defined, by induction on i, as follows:
when
, 
is defined by 
;when
, 
is the unique homomorphism of 
-algebras 
extending h (Theorem 6);- when
, 
is defined as follows:- on posets P of width lower than i,
is 
; - on sequential posets P of width i,
is 
; - on parallel posets P of width i,
is defined relying on the irreducible parallel factorization 
of P, for some 
, by: 
Proving that 
is a homomorphism of 
-algebras is routine. Furthermore, the uniqueness of 
comes from the facts that 
extends h and that A is a generating set of 
.
Decidability
Throughout this section, A denotes an alphabet. The set of terms of some signature over A is the smallest set of finite words built from A using the operations of the corresponding signature. In this section we prove the decidability of the equational theory of 
.
Let 
be the signature of 
-algebras. We start by defining the set of terms in which we are interested.
Definition 5
The set of terms 
over A is the smallest set satisfying the following conditions:
;if
then 
;if
then 
.
By equipping 
with the operations of 
, we define a structure called the term algebra 
over A. Note that 
can be considered also as the set of trees whose leaves are labeled by 
and whose internal nodes are labeled by the operations of 
where the out-degree of each internal node coincides with the arity of the corresponding operation.
Two terms 
are equivalent if 
can be derived from t using the axioms which 
satisfy (denoted 
). This equivalence relation is actually a congruence. It is well-known that 
is absolutely free i.e. it is freely generated by A in the class containing all the algebras of signature 
. In addition, as a consequence of Theorem 7, 
is isomorphic to 
(see eg. [1, Theorem 1.3.2]). This isomorphism can be defined by 
and 
for all 
.
Then we have:
Proposition 2
Let 
. Then 
if and only if 
holds in 
.
As a consequence, proving the decidability of the equational theory of 
can be reduced to decide whether 
.
Theorem 8
Let 
. It is decidable whether 
.
We now give a quick outline of the proof. The terms t and 
can be interpreted as particular forms of rational expressions over languages of 
, see [6]. By extension of a well-known result of Büchi on ordinals, it is known from [2] that a language of 
is rational if and only if it is definable in an extension, named P-MSO, of the so-called monadic second-order logic. Two P-MSO formulæ 
and 
such that 
and 
can effectively be built from t and 
. We have 
if and only if 
. Theorem 8 follows from the decidability of the P-MSO theory of 
[2, Theorem 6].
This decision procedure has a non-elementary complexity. Another proof with an exponential complexity (in the size of 
) can be derived from the proof of [12, Theorem 7.6], in which the 
-power is not considered, by replacing the use of [12, Theorem 7.3] by [9, Corollary 3.19].
Acknowledgements
We would like to thank the anonymous referees for their comments on this work. One of them pointed out that Theorem 3 can be deduced from Theorem 1 using the theory of categories, and in particular works by Fiore and Hur [13], Robinson [19], Adámek, Rosicky, Velbil et al.
Contributor Information
Nataša Jonoska, Email: jonoska@mail.usf.edu.
Dmytro Savchuk, Email: savchuk@usf.edu.
Amrane Amazigh, Email: Amazigh.Amrane@etu.univ-rouen.fr.
Nicolas Bedon, Email: Nicolas.Bedon@univ-rouen.fr.
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