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The bideterministic concatenation product
Jean-Eric Pin, Denis Thérien
To cite this version:
Jean-Eric Pin, Denis Thérien. The bideterministic concatenation product. International Journal of
Algebra and Computation, World Scientic Publishing, 1993, 3, pp.535-555. �hal-00020072�
The bideterministic concatenation product
Jean-Eric Pin and Denis Th´erien
∗
Bull Research and Development, Rue Jean-Jaur`es, 78340 Les
Clayes-sous-Bois, France
Abstract
This paper is devoted to the study of the bideterministic concatenation
product, a variant of the concatenation product. We give an algebraic char-
acterization of the varieties of languages closed under this product. More pre-
cisely, let V be a variety of monoids, V the corresponding variety of languages
and
ˆ
V the smallest variety containing V and the bideterministic products of
two languages of V. We give an algebraic description of the variety of monoids
b
V corresponding to
ˆ
V. For instance, we compute
b
V when V is one of the
following varieties : the variety of idempotent and commutative monoids, the
variety of monoids which are semilattices of groups of a given variety of groups,
the variety of R-trivial and idempotent monoids. In particular, we show that
the smallest variety of languages closed under bideterministic product and con-
taining the language {1}, corresponds to the variety of J -trivial monoids with
commuting idempotents. Similar results were known for the other variants
of the concatenation product, but the corresponding algebraic operations on
varieties of monoids were based on variants of the semidirect product and of
the Malcev product. Here the operation V →
b
V makes use of a construction
which associates to any finite monoid M an expansion
c
M , with the following
properties:
(1) M is a quotient of
c
M ,
(2) the morphism π :
c
M → M induces an isomorphism between the sub-
monoids of
c
M and of M generated by the regular elements and
(3) the inverse image under π of an idempotent of M is a 2-nilpotent semi-
group.
This paper assumes some familiarity with Eilenberg’s theory of varieties and
especially the notion of syntactic monoid of a recognizable language. References
for this theory are [6, 7, 10]. The main result of this theory states that there
exists a one-to-one correspondence between certain families of recognizable sets (the
varieties of languages) and certain families of finite semigroups (the varieties of finite
semigroups).
A fundamental result of Sch¨utzenberger [13] states that the smallest variety of
languages closed under concatenation product corresponds to the variety of ape-
riodic monoids. Since then, an important part of the existing literature on vari-
eties has been devoted to the study of the concatenation product and its variants.
These variants include the weak forms of the concatenation product introduced by
Sch¨utzenberger [14] (the unambiguous product and the left and right deterministic
products) and the counter product introduced by Straubing [15]. This paper is de-
voted to the study of the two-sided version of the deterministic products, called the
bideterministic product.
∗
Research on this paper was supported for the first author by PRC ”Math´ematiques et Infor-
matique” and for the second author by the NSERC grant no. A4546 et FCAR grant no 89-EQ-2933.
1
The general setting for this type of result can be summarized as follows. Let ◦
be a binary operation on languages — in our case the concatenation product or one
of its variants — and let V be the variety of languages corresponding to a variety
of monoids V. Denote by V
0
the smallest variety containing V and closed under ◦.
The question is to describe the varieties of monoids V
0
corresponding to V
0
. For
all the variants of concatenation mentionned previously, the variety V
0
is equal to
a Malcev product of the form W
M
V, where W is a certain variety of semigroups
[16, 8, 9, 12, 17, 18]. This variety W is given in the following table:
Product type Variety W such that V
0
= W
M
V
concatenation aperiodic semigroups
unambiguous semigroups S such that eSe = e for each idempotent e ∈ S
right deterministic semigroups S such that eS = e for each idempotent e ∈ S
left deterministic semigroups S such that Se = e for each idempotent e ∈ S
counter semigroups which are locally solvable groups
This is no longer true for the bideterministic product: in this case, the variety V
0
cannot be written as a Malcev product of some variety with V and a new algebraic
operation is required. This new operation relies on a construction of independent
interest, which associates to any monoid M a certain expansion
c
M, with the following
properties: M is a quotient of
c
M and the morphism π :
c
M → M induces an
isomorphism from hReg(
c
M)i, the submonoid of
c
M generated by the regular elements
of
c
M, onto hReg(M )i. Furthermore,the inverse image under π of an idempotent of
M is a 2-nilpotent semigroup. Our construction is somewhat reminiscent of the
expansion proposed by Birget, Margolis and Rhodes in [4, 5], but turns out to be
different, as we shall see on an example.
Now the key result states that a variety of languages is closed under bidetermin-
istic product if and only if the corresponding variety of monoids is closed under this
expansion. We also give a more precise version of this result. Let V be a variety of
languages and let V be the corresponding variety of monoids. Let
ˆ
V be the smallest
variety containing V and the bideterministic products of two languages of V. Then
the variety of monoids corresponding to
ˆ
V is the variety of monoids generated by
the monoids of the form
ˆ
M for some M ∈ V. Similar results are known for the
other variants of products, but again, there are based on totally different algebraic
constructions (essentially variants of the semidirect product).
We compute
b
V for various varities V, including the variety of idempotent and
commutative monoids, the variety of monoids which are semilattices of groups of a
given variety of groups and the variety of R-trivial and idempotent monoids.
As a byproduct, we characterize the smallest non trivial variety of languages
containing the language {1} and closed under bideterministic product : the corre-
sponding variety of monoids is the variety of J -trivial monoids whose idempotents
commute.
1 Some preliminaries.
In this section, we recall some basic definitions or facts about finite semigroups and
languages. All semigroups and monoids considered in this paper are either finite or
free, although some results could be easily extended to periodic semigroups.
Let S be a semigroup. We denote by S
1
the semigroup equal to S if S has an
identity and to S ∪{1}, where 1 is a new identity, otherwise. We denote by
E
(S) the
set of idempotents of S. For each element s of S, the subsemigroup of S generated
2
by s contains a unique idempotent, denoted s
ω
. If P is a subset of M, hP i denotes
the submonoid generated by P .
Given s, t ∈ S, we say that s is R-below t (denoted s ≤
R
t) if there exists x ∈ S
1
such that s = tx. The elements s and t are R-equivalent (denoted s R t) if s ≤
R
t
and t ≤
R
s. Finally, we denote s <
R
t if s is R-below t but is not R-equivalent
with t. The relations ≤
L
, L and <
L
are defined dually. For instance, s ≤
L
t if there
exists x ∈ S
1
such that s = xt.
Let s be a semigroup and let s be an element of S. An element ¯s of S is called a
weak inverse of s if ¯ss¯s = ¯s. It is an inverse of s if ¯ss¯s = ¯s and s¯ss = s. In this case,
s is an inverse of ¯s. An element which has an inverse is called regular. We denote
by Reg(S) the set of regular elements of a semigroup S. The following propositions
state some elementary properties of weak inverses.
Proposition 1.1 Let ¯s be a weak inverse of s. Then s¯s and ¯ss are idempotent and
¯s is an inverse of s¯ss.
Proof. If ¯s is a weak inverse of s, we have ¯ss¯s = ¯s. This implies in particular
s¯ss¯s = s¯s and ¯ss¯ss = ¯ss and thus s¯s and ¯ss are idempotent. We also have
(s¯ss)¯s(s¯ss) = (s¯s)(s¯s)(s¯s)s = s¯ss and ¯s(s¯ss)¯s = ¯s
Thus ¯s is an inverse of s¯ss.
Proposition 1.2 Let s and t be elements of a semigroup S such that s R st (resp.
ts L s). Then there exists a weak inverse
¯
t of t such that st
¯
t = s (resp.
¯
tts = s).
Proof. Since s R st, there exists an element t
0
∈ S
1
such that stt
0
= s. Let
ω be an integer such that (tt
0
)
ω
is idempotent, and set
¯
t = t
0
(tt
0
)
2ω−1
. Then
st
¯
t = stt
0
(tt
0
)
2ω−1
= s. Furthermore
¯
tt
¯
t = t
0
(tt
0
)
2ω−1
tt
0
(tt
0
)
2ω−1
= t
0
(tt
0
)
4ω−1
=
t
0
(tt
0
)
2ω−1
=
¯
t. Thus
¯
t is a weak inverse of t. The proof for the L relation is
dual.
A monoid M divides a monoid N if M is a quotient of a submonoid of N. A
variety of finite monoids is a class of finite monoids closed under taking submonoids,
quotients and finite direct products.
Recall that a relational morphism between monoids M and N is a relation τ :
M → N such that:
(1) (mτ)(nτ) ⊂ (mn)τ for all m, n ∈ M,
(2) (mτ) is non-empty for all m ∈ M ,
(3) 1 ∈ 1τ
Equivalently, τ is a relation whose graph
graph(τ) = { (m, n) | n ∈ mτ }
is a submonoid of M × N that projects onto M.
Let V and W be varieties. The Malcev product of V and W is the variety
V
M
W defined as follows
V
M
W = { M | There is a relational morphism τ : M → N with N ∈ W
and such that eτ
−1
∈ V for all idempotents e ∈ N }
Let A be a finite set, called the alphabet, whose elements are letters. We denote by
A
∗
the free monoid over A. Elements of A
∗
are words. In particular, the empty
word, denoted by 1, is the identity of A
∗
. A language is a subset of A
∗
.
3
Let M be a monoid and L be a language of A
∗
. A monoid morphism ϕ : A
∗
→ M
recognizes a language L if there exists a subset P of M such that L = ϕ
−1
(P ). The
syntactic congruence of L is the equivalence ∼
L
on A
∗
defined by
u ∼
L
v if and only if, for every x, y ∈ A
∗
(xuy ∈ L ⇐⇒ xvy ∈ L).
The quotient A
∗
/∼
L
is the syntactic monoid of L and the natural morphism η :
A
∗
→ M (L) is called the syntactic morphism: it recognizes L and every surjective
morphism ϕ : A
∗
→ M that recognizes L can be factorized through it, that is, there
is a surjective morphism α : M → M(L) such that η = α ◦ ϕ.
For technical reasons, it is more appropriate to use a variant of the concatenation
product called the marked product. The results stated in the introduction refer to
this product. Given a finite alphabet A and a letter a of A, the marked product of
two subsets (also called languages) L
0
and L
1
of the free monoid A
∗
is the language
L
0
aL
1
= {u ∈ A
∗
| u = u
0
au
1
for some u
0
∈ L
0
and u
1
∈ L
1
}
Unambiguous, left and right deterministic products were introduced by Sch¨utzen-
berger. A product L = L
0
aL
1
is unambiguous if every word u of L has a unique
decomposition of the form u = u
0
au
1
with u
0
∈ L
0
and u
1
∈ L
1
. It is left deter-
ministic if every word of L has exactly one prefix in L
0
a. This means that in order
to find the decomposition u = u
0
au
1
of a word of L, it suffices to read u from left
to right: the first prefix of u in L
0
a will give u
0
a, and thus the decomposition. Du-
ally, a product L = L
0
aL
1
is right deterministic if every word of L has exactly one
suffix in aL
1
. A product is called bideterministic if it is both deterministic and an-
tideterministic. Sch¨utzenberger [14] characterized the smallest variety of languages
containing the language {1} and closed under unambiguous (resp. deterministic,
antideterministic) products. Later on, it was shown in [8, 9, 12] that the closure of
a variety of languages under unambiguous (resp. left deterministic, right determin-
istic) product correspond to the Malcev product V → LI
M
V (resp. V → K
M
V,
V → K
r
M
V), where LI, K and K
r
are respectively the varieties of semigroups S
such that, for every idempotent e ∈ S, eSe = e, (resp. eS = e, Se = e).
2 An expansion.
In this section, we give the formal definition of our new expansion, which is related
to certain special factorizations of words.
Let M be a monoid, and let ϕ : A
∗
→ M be a surjective (monoid) morphism.
A good factorization (with respect to ϕ) is a triple (x
0
, a, x
1
) ∈ A
∗
× A × A
∗
such
that ϕ(x
0
a) <
R
ϕ(x
0
) and ϕ(ax
1
) <
L
ϕ(x
1
). A good factorization of a word x ∈ A
∗
is a good factorization (x
0
, a, x
1
) such that x = x
0
ax
1
. Two good factorizations
(x
0
, a, x
1
) and (y
0
, b, y
1
) are equivalent if ϕ(x
0
) = ϕ(y
0
), ϕ(x
1
) = ϕ(y
1
) and a = b.
In particular, this implies ϕ(x
0
ax
1
) = ϕ(y
0
by
1
). Here is a first useful lemma.
Lemma 2.1 Let (x
0
, a, x
1
) be a good factorization, let x
0
0
be a right factor of x
0
and
let x
0
1
be a left factor of x
1
. Then (x
0
0
, a, x
0
1
) is a good factorization.
Proof. Set x
0
= x
00
0
x
0
0
and x
1
= x
0
1
x
00
1
. If (x
0
0
, a, x
0
1
) is not a good factorization,
then ϕ(x
0
0
a) R ϕ(x
0
0
) or ϕ(ax
0
1
) L ϕ(x
0
1
). We treat the first case, but the other case
is dual. Since R is stable on the left, we have ϕ(x
00
0
)ϕ(x
0
0
a) R ϕ(x
00
0
)ϕ(x
0
0
), whence
ϕ(x
0
) R ϕ(x
0
a), a contradiction, since (x
0
, a, x
1
) is a good factorization.
Define a relation ∼
ϕ
on A
∗
, by setting, for each x, y ∈ A
∗
, x ∼
ϕ
y if and only if the
following three conditions are satisfied:
4