In this work we illustrate how the study of the topological semigroups of implicit operations of a pseudovariety V  can be useful to the knowledge of V. In particular, for  subpseudovarieties of  DS, we find factorizations of the implicit operations in terms of regular and explicit ("words'') ones.
For a better knowledge of V,  we try to solve some kind of "word problem'' on V i.e., decide when two implicit operations  are the same in V. Such study has already been done for some pseudovarieties, namely J, R and L. As an application of this study we calculate some joins of pseudovarieties

Azevedo, Assis

Universidade do Minho: RepositoriUM

Implicit operations on DS

Implicit operations on DSAssis AzevedoUniversity of MinhoBraga, PortugalAbstractIn this work we illustrate how the study of the topological semigroups ΩnV (fora pseudovariety V) can be useful to the knowledge of V. In particular, for subpseu-dovarieties of DS, we find factorizations of implicit operations in terms of regularand explicit (“words”) ones. For a better knowledge of V, we try to solve some kindof “word problem” on V i.e., decide when two implicit operations are the same in V.Such study has already been done for some pseudovarieties, namely J [4], R and L[7].As an application of the study of implicit operations, we calculate some joins ofpseudovarieties.1 IntroductionA pseudovariety of semigroups is a non-empty class of finite semigoups closed under theformation of homomorphic images, subsemigroups and finite products.Lawvere [12] defined a n-ary implicit operation on a pseudovariety V as a new n-aryoperation that commutes with all homomorphisms between elements of V. The semigroupof the n-ary implicit operations on V is denoted by ΩnV.Reiterman [13] proved, using topological methods, that pseudovarieties can be definedby pseudoidentities (formal identities of implicit operations).Reiterman’s Theorem is a very useful tool for the resolution of some questions aboutpseudovarieties. For instance, the calculation of joins of pseudovarieties. Some resultsabout joins can be very surprising. In fact, the join of very simple decidable pseudovarietiescan have an undecidable membership problem [1]. In this work we show how Reiterman’sTheorem can be used to calculate some joins of pseudovarieties contained in DS, thepseudovariety of the semigroups such that all its regular J -classes are subsemigroups.2 PreliminariesLet V be a pseudovariety.Definition 2.1 For n ∈ IN , an n-ary implicit operation on V is given by pi = (piS)S∈Vwhere for each S ∈ V, piS : Sn → S is a function, and for each homomorphism ϕ : S → Twith S, T ∈ V, ϕ ◦ piS = piT ◦ ϕ(n).The set of all n-ary implicit operations on V is denoted by ΩnV. ΩnV has a naturalstructure of semigroup: for pi, ρ ∈ ΩnV and S ∈ V, we define (piρ)S = piSρS.The subsemigroup of ΩnV generated by {x1, . . . , xn} is denoted by ΩnV (where(xi)S(a1, . . . , an) = ai for i ∈ {1, . . . , n}, S ∈ V and a1, . . . , an ∈ S). The elementsof ΩnV are called explicit operations.Another example of an implicit operation is given by the unary operation x 7→ xωsuch that, for a finite semigroup S and a ∈ S, aω is the idempotent of the subsemigroupgenerated by a. See [6] for the determination of all unary implicit operation.For a set Σ of pseudoidentities for V, we consider the subclass of V (which is in facta pseudovariety) defined by Σ as being[[Σ]]V = {A ∈ V : A |= Σ} = {A ∈ V : ∀ pi = ρ ∈ Σ, piA = ρA}.Reiterman [13] proved that every pseudovariety is of this form. The reader is referredto Reiterman [13] and Almeida [3] and [5] for general results on ΩnV. In general, we write[[Σ]] instead of [[Σ]]S, where S denotes the class of all finite semigroups.The following list of pseudovarieties will appear in the sequel.Com = {commutative semigroups} =[[xy = yx]],G = {groups} =[[xωy = yxω = y]],Ab = {abelian groups} =[[xy = yx, xωy = y]],R = {R-trivial semigroups} =[[xω+1 = xω, (xy)ωx = (xy)ω]],L = {L-trivial semigroups} =[[xω+1 = xω, y(xy)ω = (yx)ω]],J = {J -trivial semigroups} =[[xω+1 = xω, (xy)ω = (yx)ω]],K = {semigroups such that idempotents are left zeros} =[[xωy = xω]],D = {semigroups such that idempotents are right zeros} =[[yxω = xω]],Perm = {permutative semigroups} =[[xωyzxω = xωzyxω]].Reiterman [13] introduced a topology on ΩnV (n ∈ IN) such that it becomes a compactmetric totally disconected topological semigroup, admiting ΩnV as a dense subspace andsuch that its finite (and discrete) continuous homomorphic images are the n-generatedelements of V.The convergence of a sequence (pim)m∈IN in ΩnV to pi is equivalent to the followingcondition,(∀k ∈ IN)(∀S ∈ V)(∃p ∈ IN) [m ≥ p, |S| ≤ k ⇒ S |= pim = pi] . (1)For a non empty finite set A we denote by A+ the free semigroup generated by A. Asthe class S satisfies no nontrivial identity, ΩnS (n = |A|) can be identified with A+.3 Factorization of implicit operations on DSIn this section, we prove that every implicit operation on a subpseudovariety V of DS,the class of finite semigroups such that its regular J -classes are subsemigroups, is a finiteproduct of regular and explicit operations. First we need to caracterize the regular implicitoperations on V. More precisely, given a sequence in ΩnV with limit pi in ΩnV, we wantconditions determining when pi is regular.The case of the unary implicit operations is very easy to study, because every nonexplicit operation is regular [6].Definition 3.1 For a pseudovariety V, DV is the class of finite semigroups such that itsregular J -classes are subsemigroups of V.Note 3.2 The class DS can also be defined as the class of finite semigroups whose regularelements are group elements, or the class of semigroups S such that, for all a ∈ S, theset {b ∈ S : a ≤J b} is a subsemigroup of S. DS can be defined by the pseudoidentity[(xy)ω(yx)ω(xy)ω]ω = (xy)ω or by [(xy)ω+1x]ω+1 = (xy)ω+1x [10].For a pseudovariety V and a implicit operation pi = pi(x1, . . . , xn) ∈ ΩnV we say thatpi depends on xi if there exists S ∈ V such that piS depends on the ith-component. Also,we define c(pi) as the set of such xi. For pi ∈ ΩnV, c(pi) is said to be the content of pi.Lemma 3.3 [10] If V is a pseudovariety containing Sl thenc : ΩnV −→ P{x1, . . . , xn}is a continuous homomorphism (where P{x1, . . . , xn}, the set of subsets of {x1, . . . , xn},is considered as a discrete topological semigroup under union).Using (1) together with the homomorphism c, we obtain some characterizations of DS.Lemma 3.4 i) If pi, ρ are implicit operations on DS such that c(ρ) ⊆ c(pi) then(piωρωpiω)ω = piω.ii) DS is the greatest pseudovariety V, such that, for any regular implicit operationspi, ρ ∈ ΩnV, pi and ρ are J -equivalent if and only if they have the same content.ii) For a subpseudovariety of DS containing Sl, ΩnV has 2n − 1 regular J -classes.¤For the characterization of the regular implicit operations on DS, we make use of thefollowing definition.Definition 3.5 Let V be a pseudovariety such that ΩnV is equal, as a semigroup, to{x1, . . . , xn}+. For w, u ∈ ΩnV define,[wu]= max{r : ur is a subword of w}.Lemma 3.6 [4] If V is a pseudovariety containing J and u ∈ ΩnV, then the mappingw 7→ [wu ] (w ∈ ΩnV) is uniformly continuous, and so it extends to a continuous mappingΩnV −→ IN ∪ {∞}.pi 7→ [wu ]The following lemma is crucial for the characterization of regular implicit operationson DS.Lemma 3.7 If S is a semigroup of DS and w and u are words with the same contentsuch that[wu]> |S|, then w assumes only regular values in S, that is S |= w = wω+1.Proof: Let c(w) = {x1, . . . , xn} and w = u1 · · ·uk, where u1, . . . , uk are words such thatk =[wu]and[uiu]= 1.Let bp = u1 · · ·up with p ≤ k. For a1, . . . , an ∈ S, as |S| < k, there exist r, s ∈ INsuch that r < s ≤ k and br(a1, . . . , an) = bs(a1, . . . , an). If, for simplicity, we denoteui(a1, . . . , an) by ui we have,brur+1 · · ·us = bsur+1 · · ·us= br(ur+1 · · ·us)2 = · · · = br(ur+1 · · ·us)ω+1,and so,w = br(ur+1 · · ·us)ω+1 · · ·uk=[br(ur+1 · · ·us)ω+1 · · ·uk]ω+1= wω+1.¤As a consequence of this Lemma, we have the following corollary.Corollary 3.8 If V is a subpseudovariety of DS containing J and pi ∈ ΩnV, then pi isregular if and only if, for any word u,[piu] ∈ {0,∞}.¤For the factorization of implicit operations on DS, in terms of explicit and regularelements, we need the following lema which is an easy generalization of a similar resultproved in [4] for J.Lemma 3.9 For A = {x1, . . . , xn} let B be the set of all u ∈ A+ without repeated letterssuch that c(u) = A. Then the mappingϕ : ΩnDS −→ IN0.pi 7→ |{u ∈ B : [piu ] 6∈ {0,∞}} |has the following properties:i) if ϕ(pi) = 0 then pi is regular;ii) if ϕ(pi) 6= 0 then, for u ∈ B such that [piu ] = r 6∈ {0,∞}, there exists a sequence(vk)k∈IN in ΩnDS with limit pi, such that• vk = vk,0a1 · · · vk,s−1asvk,s where ur = a1 · · · as and ai 6∈ c(vk,i−1);• limkvk,i = pii;• ϕ(pii) < ϕ(pi).Then, using induction we have the following theorem.Theorem 3.10 If pi ∈ ΩnV (V ⊆ DS), then pi can be decomposed as pi = u0pi1u1 · · ·pikuk(k ∈ IN) such that• ui ∈ {x1, . . . , xn}∗ and pii is regular,• “the last letter of ui” is not in c(pii+1) and “the first letter of ui” is not in c(pii),• if ui = 1, then c(pii) and c(pii+1) are incomparable under inclusion.¤For DO, the class of semigroups of DS such that the regular J -classes are orthodoxsubsemigroups, we know more about regular implicit operations. We begin by a lemma.Lemma 3.11 Let S ∈ DO and e, a, b ∈ S such that e = e2 and e ≤J a, b. Theneabe = eaebe.Proof: As ea and be are group elements (since they lie in the J -class of e) we have,eabe = ea.(ea)ω(be)ω.be.As S is orthodox, (ea)ω(be)ω is an idempotent. But, as (ea)ω(be)ω is H-equivalent to e,(ea)ω(be)ω = e.¤Theorem 3.12 If V is a subpseudovariety of DO, then every regular implicit operationon V is determined by its restriction to V ∩G.Proof: Let pi ∈ ΩnV be regular (pi = piω+1) and (um)m∈IN a sequence in ΩnV with limitpi. For S ∈ V let, according to (1), k ∈ IN be such that S |= pi = uk.For a1, . . . , an ∈ S and e = (pi)ωS(a1, . . . , an), let H be the H-class of e (H ∈ V ∩G).ThenpiS(a1, . . . , an) = uk(a1, . . . , an)= euk(a1, . . . , an)e= uk(ea1e, . . . , eane), by Lemma 3.11= piH(ea1e, . . . , eane).¤At the level of J, the “word problem” has been solved as follows.Theorem 3.13 [4] Let pi, ρ ∈ ΩnJ. Then:i) J |= pi = ρ if and only if, for all u ∈ ΩnJ,[piu]=[ρu];ii) if pi = u0pi1u1 · · ·pikuk and ρ = u0ρ1v1 · · · ρsvs are factorizations as in Theorem3.10, then J |= pi = ρ if and only if k = s, ui = vi (i = 0, . . . , k) and c(pii) = c(ρi)(i = 1, . . . , k).Moreover, ΩnJ is the free semigroup with a unary operation (x 7→ xω) over the set{x1, . . . , xn} in the variety defined by the identities (xy)ω = (yx)ω = (xωyω)ω and xωx =xxω = xω = (xω)ω.As a consequence of this Theorem and Theorem 3.10, we have the following result,where J ∗ denotes the congruence generated by J .Corollary 3.14 If U is a subpseudovariety of DS containing J, thenU |= piJ ∗ρ⇐⇒ J |= pi = ρ.4 ApplicationsThe decomposition of an implicit operation, stated in Theorem 3.10, can be improved forsome pseudovarieties. Such decompositions were fulcral on the calculations of some joinsof pseudovarieties.Examples 4.1 1) Let ZE = {semigroups such that the idempotents are central} =[[xωy = yxω]].Then, every implicit operation on ZE is of the form pi = u0pi1u1 · · ·pikuk with• ui explicit operations,• pii regular operations,• c(pi1) = · · · = c(pik),• c(uo · · ·uk) ∩ c(pii) = ∅.This decomposition was first obtained [2] independently of Theorem 3.10.2) Let V = [[(yx)ωy(zy)ω = (yx)ω(zy)ω]]. Every implicit operation on V is of the formpi1 · · ·pik with• pii is explicit or idempotent,• “last letter of pii” is not in c(pii+1),• “first letter of pii” is not in c(pii−1).3) Let V = DPerm ∩ [[xωyxω+1 = xω+1yxω]], where Perm is the class of semigroupssatisfying a nontrivial permutative identity. Every implicit operation on V is of the formpi = u0pi1u1 · · ·pikuk as in DS, with• pii = piωi piipiωi ,• pii a product of regular unary implicit operation,• c(pii) ⊆ c(pii) \ ∪j<ic(pij).Using these decompositions we obtain the following theorems.Theorem 4.2 [2] G ∨Com = ZE.Theorem 4.3 [7] R ∨ L = [[(yx)ωy(zy)ω = (yx)ω(zy)ω]].Theorem 4.4 [9] The sublattice of the subpseudovarieties of S generated by J, K, D,Com and Ab is´´´´´CCCCCCCCCCCCCCCCCC´´´´´»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»CCCCCCCCCCCCCCCCCCD»»»»»»»»»»»»»»»»»»»»»»»»!!!!!!!!´´´´´CCCCCCCCCCCCCCCCCC´´´´´CCCCCCCCCCCCCCCCCCCCCCCCCCC!!!!!!!!Ab!!!!Com´´´´´CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC´´´´´»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»´´´´´´´´´´CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC´´´´´J´´´´´»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»»K»»»»»»»»»»»»ttt tttttttttttt tttttttt ttand all these pseudovarieties are decidable.This last result is a consequence of the following theorem.Theorem 4.5 [9] i) Let V be a pseudovariety such that W ⊆ V ⊆ Com∨W, where Wis equal to I, K, D or LI. Then J ∨V is the intersection of DV with[[xωyxω+1 = xω+1yxω, γ1 =δ1, (αβ)ω+1 =β(αβ)ω, (εδ)ω+1 =(εδ)ωε]]whereα = (ztωx)ω, ε = (xtωz)ω,β = (ztωy)ω, δ = (ytωz)ω,γ1 =xωa(st)ωbyω if LI ⊆ V ⊆ Perm = Com ∨ LIxωa(st)ω if K ⊆ V ⊆ [[xωyz = xωzy]] = Com ∨K(st)ωbyω if D ⊆ V ⊆ [[yzxω = zyxω]] = Com ∨D1 if V ⊆ Comandδ1 =xωa(ts)ωbyω if LI ⊆ V ⊆ Permxωa(ts)ω if K ⊆ V ⊆ [[xωyz = xωzy]] = Com ∨K(ts)ωbyω if D ⊆ V ⊆ [[yzxω = zyxω]] = Com ∨D1 if V ⊆ Com.ii) If V is a pseudovariety such that N ⊆ V ⊆ Com∨N then J∨V = J∨ (V∩Com).iii) If V is a pseudovariety satisfying the conditions of i) or ii) then J∨V is decidableif and only if V is decidable.Remark 4.6 In all these examples we use a specific decomposition of implicit operations.Let V ∗W denote the pseudovariety generated by the (‘left unitary’) semidirect products[11] of elements of V with elements of W. We can prove that Com ∗Dk, where Dk isthe class of semigroups of D such that any product of k elements is an idempotent, is apseudovariety not contained in DS with the property that every implicit operation is afinite product of regular and explicit operations [8]. However Com ∗V is an example ofa pseudovariety not satisfying this last property [8].So, the following question arises: which are the pseudovarieties such that every implicitoperation is a finite product of regular and explicit operations?References[1] D. Albert, R. Baldinger e J. Rhodes, Undecidibility of the identity problem for finitesemigroups with applications, preprint, 1986.[2] J. Almeida, Some pseudovariety joins involving the pseudovariety of finite groups,Semigroup Forum 37 (1988) 53-57.[3] , The algebra of implicit operations, Algebra Universalis 26 (1989), 16-72.[4] , Implicit operations on finite J -trivial semigroups and a conjecture of I.Simon, to appear in J. Pure Appl. Algebra.[5] , On pseudovarieties, varieties of languages, filters of congruences,pseudoidentities and related topics, to appear in Algebra Universalis.[6] J. Almeida e A. Azevedo, Implicit operations on certain classes of semigroups, Semi-groups and their Applications by S. Goberstein and P.Higgins (Eds.), Proceedings1986 Chico Conf., D. Reidel, 1987 (1-11).[7] , The join of the pseudovarieties of R-trivial and L-trivial monoids,J. Pure Appl. Algebra 60 (1989) 129-137.[8] , On regular implicit operations, in preparation.[9] A. Azevedo, The join of the pseudovariety J with permutative pseudovarieties, toappear in Proceedings Conferência International de Álgebra, Lisboa, 1988.[10] , Operações Implícitas sobre Pseudovariedades de Semigrupos, Aplicações,Doctoral dissertation, University of Porto, 1989.[11] S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York,1976.[12] F. W. Lawvere, Functional semantics of algebraic theories, Doctoral dissertation,University of Columbia, New York, 1963.[13] J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982)1-10.

Implicit operations on DS
Assis Azevedo
University of Minho
Braga, Portugal
Abstract
In this work we illustrate how the study of the topological semigroups ΩnV (for
a pseudovariety V) can be useful to the knowledge of V. In particular, for subpseu-
dovarieties of DS, we find factorizations of implicit operations in terms of regular
and explicit (“words”) ones. For a better knowledge of V, we try to solve some kind
of “word problem” on V i.e., decide when two implicit operations are the same in V.
Such study has already been done for some pseudovarieties, namely J [4], R and L
[7].
As an application of the study of implicit operations, we calculate some joins of
pseudovarieties.
1 Introduction
A pseudovariety of semigroups is a non-empty class of finite semigoups closed under the
formation of homomorphic images, subsemigroups and finite products.
Lawvere [12] defined a n-ary implicit operation on a pseudovariety V as a new n-ary
operation that commutes with all homomorphisms between elements of V. The semigroup
of the n-ary implicit operations on V is denoted by ΩnV.
Reiterman [13] proved, using topological methods, that pseudovarieties can be defined
by pseudoidentities (formal identities of implicit operations).
Reiterman’s Theorem is a very useful tool for the resolution of some questions about
pseudovarieties. For instance, the calculation of joins of pseudovarieties. Some results
about joins can be very surprising. In fact, the join of very simple decidable pseudovarieties
can have an undecidable membership problem [1]. In this work we show how Reiterman’s
Theorem can be used to calculate some joins of pseudovarieties contained in DS, the
pseudovariety of the semigroups such that all its regular J -classes are subsemigroups.
2 Preliminaries
Let V be a pseudovariety.
Definition 2.1 For n ∈ IN , an n-ary implicit operation on V is given by π = (πS)S∈V
where for each S ∈ V, πS : Sn → S is a function, and for each homomorphism ϕ : S → T
with S, T ∈ V, ϕ ◦ πS = πT ◦ ϕ(n).
The set of all n-ary implicit operations on V is denoted by ΩnV. ΩnV has a natural
structure of semigroup: for π, ρ ∈ ΩnV and S ∈ V, we define (πρ)S = πSρS.
The subsemigroup of ΩnV generated by {x1, . . . , xn} is denoted by ΩnV (where
(xi)S(a1, . . . , an) = ai for i ∈ {1, . . . , n}, S ∈ V and a1, . . . , an ∈ S). The elements
of ΩnV are called explicit operations.
Another example of an implicit operation is given by the unary operation x 7→ xω
such that, for a finite semigroup S and a ∈ S, aω is the idempotent of the subsemigroup
generated by a. See [6] for the determination of all unary implicit operation.
For a set Σ of pseudoidentities for V, we consider the subclass of V (which is in fact
a pseudovariety) defined by Σ as being
[[Σ]]V = {A ∈ V : A |= Σ} = {A ∈ V : ∀ π = ρ ∈ Σ, πA = ρA}.
Reiterman [13] proved that every pseudovariety is of this form. The reader is referred
to Reiterman [13] and Almeida [3] and [5] for general results on ΩnV. In general, we write
[[Σ]] instead of [[Σ]]S, where S denotes the class of all finite semigroups.
The following list of pseudovarieties will appear in the sequel.
Com = {commutative semigroups} =[[xy = yx]],
G = {groups} =[[xωy = yxω = y]],
Ab = {abelian groups} =[[xy = yx, xωy = y]],
R = {R-trivial semigroups} =[[xω+1 = xω, (xy)ωx = (xy)ω]],
L = {L-trivial semigroups} =[[xω+1 = xω, y(xy)ω = (yx)ω]],
J = {J -trivial semigroups} =[[xω+1 = xω, (xy)ω = (yx)ω]],
K = {semigroups such that idempotents are left zeros} =[[xωy = xω]],
D = {semigroups such that idempotents are right zeros} =[[yxω = xω]],
Perm = {permutative semigroups} =[[xωyzxω = xωzyxω]].
Reiterman [13] introduced a topology on ΩnV (n ∈ IN) such that it becomes a compact
metric totally disconected topological semigroup, admiting ΩnV as a dense subspace and
such that its finite (and discrete) continuous homomorphic images are the n-generated
elements of V.
The convergence of a sequence (πm)m∈IN in ΩnV to π is equivalent to the following
condition,
(∀k ∈ IN)(∀S ∈ V)(∃p ∈ IN) [m ≥ p, |S| ≤ k ⇒ S |= πm = π] . (1)
For a non empty finite set A we denote by A+ the free semigroup generated by A. As
the class S satisfies no nontrivial identity, ΩnS (n = |A|) can be identified with A+.
3 Factorization of implicit operations on DS
In this section, we prove that every implicit operation on a subpseudovariety V of DS,
the class of finite semigroups such that its regular J -classes are subsemigroups, is a finite
product of regular and explicit operations. First we need to caracterize the regular implicit
operations on V. More precisely, given a sequence in ΩnV with limit π in ΩnV, we want
conditions determining when π is regular.
The case of the unary implicit operations is very easy to study, because every non
explicit operation is regular [6].
Definition 3.1 For a pseudovariety V, DV is the class of finite semigroups such that its
regular J -classes are subsemigroups of V.
Note 3.2 The class DS can also be defined as the class of finite semigroups whose regular
elements are group elements, or the class of semigroups S such that, for all a ∈ S, the
set {b ∈ S : a ≤J b} is a subsemigroup of S. DS can be defined by the pseudoidentity
[(xy)ω(yx)ω(xy)ω]ω = (xy)ω or by [(xy)ω+1x]ω+1 = (xy)ω+1x [10].
For a pseudovariety V and a implicit operation π = π(x1, . . . , xn) ∈ ΩnV we say that
π depends on xi if there exists S ∈ V such that πS depends on the ith-component. Also,
we define c(π) as the set of such xi. For π ∈ ΩnV, c(π) is said to be the content of π.
Lemma 3.3 [10] If V is a pseudovariety containing Sl then
c : ΩnV −→ P{x1, . . . , xn}
is a continuous homomorphism (where P{x1, . . . , xn}, the set of subsets of {x1, . . . , xn},
is considered as a discrete topological semigroup under union).
Using (1) together with the homomorphism c, we obtain some characterizations of DS.
Lemma 3.4 i) If π, ρ are implicit operations on DS such that c(ρ) ⊆ c(π) then
(πωρωπω)ω = πω.
ii) DS is the greatest pseudovariety V, such that, for any regular implicit operations
π, ρ ∈ ΩnV, π and ρ are J -equivalent if and only if they have the same content.
ii) For a subpseudovariety of DS containing Sl, ΩnV has 2n − 1 regular J -classes.¤
For the characterization of the regular implicit operations on DS, we make use of the
following definition.
Definition 3.5 Let V be a pseudovariety such that ΩnV is equal, as a semigroup, to
{x1, . . . , xn}+. For w, u ∈ ΩnV define,
[w
u
]
= max{r : ur is a subword of w}.
Lemma 3.6 [4] If V is a pseudovariety containing J and u ∈ ΩnV, then the mapping
w 7→ [wu
]
(w ∈ ΩnV) is uniformly continuous, and so it extends to a continuous mapping
ΩnV −→ IN ∪ {∞}.
π 7→ [wu
]
The following lemma is crucial for the characterization of regular implicit operations
on DS.
Lemma 3.7 If S is a semigroup of DS and w and u are words with the same content
such that
[
w
u
]
> |S|, then w assumes only regular values in S, that is S |= w = wω+1.
Proof: Let c(w) = {x1, . . . , xn} and w = u1 · · ·uk, where u1, . . . , uk are words such that
k =
[
w
u
]
and
[
ui
u
]
= 1.
Let bp = u1 · · ·up with p ≤ k. For a1, . . . , an ∈ S, as |S| < k, there exist r, s ∈ IN
such that r < s ≤ k and br(a1, . . . , an) = bs(a1, . . . , an). If, for simplicity, we denote
ui(a1, . . . , an) by ui we have,
brur+1 · · ·us = bsur+1 · · ·us
= br(ur+1 · · ·us)2 = · · · = br(ur+1 · · ·us)ω+1,
and so,
w = br(ur+1 · · ·us)ω+1 · · ·uk
=
[
br(ur+1 · · ·us)ω+1 · · ·uk
]ω+1
= wω+1.¤
As a consequence of this Lemma, we have the following corollary.
Corollary 3.8 If V is a subpseudovariety of DS containing J and π ∈ ΩnV, then π is
regular if and only if, for any word u,
[
π
u
] ∈ {0,∞}.¤
For the factorization of implicit operations on DS, in terms of explicit and regular
elements, we need the following lema which is an easy generalization of a similar result
proved in [4] for J.
Lemma 3.9 For A = {x1, . . . , xn} let B be the set of all u ∈ A+ without repeated letters
such that c(u) = A. Then the mapping
ϕ : ΩnDS −→ IN0.
π 7→ |{u ∈ B : [πu
] 6∈ {0,∞}} |
has the following properties:
i) if ϕ(π) = 0 then π is regular;
ii) if ϕ(π) 6= 0 then, for u ∈ B such that [πu
]
= r 6∈ {0,∞}, there exists a sequence
(vk)k∈IN in ΩnDS with limit π, such that
• vk = vk,0a1 · · · vk,s−1asvk,s where ur = a1 · · · as and ai 6∈ c(vk,i−1);
• lim
k
vk,i = πi;
• ϕ(πi) < ϕ(π).
Then, using induction we have the following theorem.
Theorem 3.10 If π ∈ ΩnV (V ⊆ DS), then π can be decomposed as π = u0π1u1 · · ·πkuk
(k ∈ IN) such that
• ui ∈ {x1, . . . , xn}∗ and πi is regular,
• “the last letter of ui” is not in c(πi+1) and “the first letter of ui” is not in c(πi),
• if ui = 1, then c(πi) and c(πi+1) are incomparable under inclusion.¤
For DO, the class of semigroups of DS such that the regular J -classes are orthodox
subsemigroups, we know more about regular implicit operations. We begin by a lemma.
Lemma 3.11 Let S ∈ DO and e, a, b ∈ S such that e = e2 and e ≤J a, b. Then
eabe = eaebe.
Proof: As ea and be are group elements (since they lie in the J -class of e) we have,
eabe = ea.(ea)ω(be)ω.be.
As S is orthodox, (ea)ω(be)ω is an idempotent. But, as (ea)ω(be)ω is H-equivalent to e,
(ea)ω(be)ω = e.¤
Theorem 3.12 If V is a subpseudovariety of DO, then every regular implicit operation
on V is determined by its restriction to V ∩G.
Proof: Let π ∈ ΩnV be regular (π = πω+1) and (um)m∈IN a sequence in ΩnV with limit
π. For S ∈ V let, according to (1), k ∈ IN be such that S |= π = uk.
For a1, . . . , an ∈ S and e = (π)ωS(a1, . . . , an), let H be the H-class of e (H ∈ V ∩G).
Then
πS(a1, . . . , an) = uk(a1, . . . , an)
= euk(a1, . . . , an)e
= uk(ea1e, . . . , eane), by Lemma 3.11
= πH(ea1e, . . . , eane).¤
At the level of J, the “word problem” has been solved as follows.
Theorem 3.13 [4] Let π, ρ ∈ ΩnJ. Then:
i) J |= π = ρ if and only if, for all u ∈ ΩnJ,
[
π
u
]
=
[
ρ
u
]
;
ii) if π = u0π1u1 · · ·πkuk and ρ = u0ρ1v1 · · · ρsvs are factorizations as in Theorem
3.10, then J |= π = ρ if and only if k = s, ui = vi (i = 0, . . . , k) and c(πi) = c(ρi)
(i = 1, . . . , k).
Moreover, ΩnJ is the free semigroup with a unary operation (x 7→ xω) over the set
{x1, . . . , xn} in the variety defined by the identities (xy)ω = (yx)ω = (xωyω)ω and xωx =
xxω = xω = (xω)ω.
As a consequence of this Theorem and Theorem 3.10, we have the following result,
where J ∗ denotes the congruence generated by J .
Corollary 3.14 If U is a subpseudovariety of DS containing J, then
U |= πJ ∗ρ ⇐⇒ J |= π = ρ.
4 Applications
The decomposition of an implicit operation, stated in Theorem 3.10, can be improved for
some pseudovarieties. Such decompositions were fulcral on the calculations of some joins
of pseudovarieties.
Examples 4.1 1) Let ZE = {semigroups such that the idempotents are central} =
[[xωy = yxω]].
Then, every implicit operation on ZE is of the form π = u0π1u1 · · ·πkuk with
• ui explicit operations,
• πi regular operations,
• c(π1) = · · · = c(πk),
• c(uo · · ·uk) ∩ c(πi) = ∅.
This decomposition was first obtained [2] independently of Theorem 3.10.
2) Let V = [[(yx)ωy(zy)ω = (yx)ω(zy)ω]]. Every implicit operation on V is of the form
π1 · · ·πk with
• πi is explicit or idempotent,
• “last letter of πi” is not in c(πi+1),
• “first letter of πi” is not in c(πi−1).
3) Let V = DPerm ∩ [[xωyxω+1 = xω+1yxω]], where Perm is the class of semigroups
satisfying a nontrivial permutative identity. Every implicit operation on V is of the form
π = u0π1u1 · · ·πkuk as in DS, with
• πi = πωi π̃iπωi ,
• π̃i a product of regular unary implicit operation,
• c(π̃i) ⊆ c(πi) \ ∪j<ic(πj).
Using these decompositions we obtain the following theorems.
Theorem 4.2 [2] G ∨Com = ZE.
Theorem 4.3 [7] R ∨ L = [[(yx)ωy(zy)ω = (yx)ω(zy)ω]].
Theorem 4.4 [9] The sublattice of the subpseudovarieties of S generated by J, K, D,
Com and Ab is
´
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and all these pseudovarieties are decidable.
This last result is a consequence of the following theorem.
Theorem 4.5 [9] i) Let V be a pseudovariety such that W ⊆ V ⊆ Com∨W, where W
is equal to I, K, D or LI. Then J ∨V is the intersection of DV with
[[xωyxω+1 = xω+1yxω, γ1 =δ1, (αβ)ω+1 =β(αβ)ω, (εδ)ω+1 =(εδ)ωε]]
where
α = (ztωx)ω, ε = (xtωz)ω,
β = (ztωy)ω, δ = (ytωz)ω,
γ1 =



xωa(st)ωbyω if LI ⊆ V ⊆ Perm = Com ∨ LI
xωa(st)ω if K ⊆ V ⊆ [[xωyz = xωzy]] = Com ∨K
(st)ωbyω if D ⊆ V ⊆ [[yzxω = zyxω]] = Com ∨D
1 if V ⊆ Com
and
δ1 =



xωa(ts)ωbyω if LI ⊆ V ⊆ Perm
xωa(ts)ω if K ⊆ V ⊆ [[xωyz = xωzy]] = Com ∨K
(ts)ωbyω if D ⊆ V ⊆ [[yzxω = zyxω]] = Com ∨D
1 if V ⊆ Com.
ii) If V is a pseudovariety such that N ⊆ V ⊆ Com∨N then J∨V = J∨ (V∩Com).
iii) If V is a pseudovariety satisfying the conditions of i) or ii) then J∨V is decidable
if and only if V is decidable.
Remark 4.6 In all these examples we use a specific decomposition of implicit operations.
Let V ∗W denote the pseudovariety generated by the (‘left unitary’) semidirect products
[11] of elements of V with elements of W. We can prove that Com ∗Dk, where Dk is
the class of semigroups of D such that any product of k elements is an idempotent, is a
pseudovariety not contained in DS with the property that every implicit operation is a
finite product of regular and explicit operations [8]. However Com ∗V is an example of
a pseudovariety not satisfying this last property [8].
So, the following question arises: which are the pseudovarieties such that every implicit
operation is a finite product of regular and explicit operations?
References
[1] D. Albert, R. Baldinger e J. Rhodes, Undecidibility of the identity problem for finite
semigroups with applications, preprint, 1986.
[2] J. Almeida, Some pseudovariety joins involving the pseudovariety of finite groups,
Semigroup Forum 37 (1988) 53-57.
[3] , The algebra of implicit operations, Algebra Universalis 26 (1989), 16-72.
[4] , Implicit operations on finite J -trivial semigroups and a conjecture of I.
Simon, to appear in J. Pure Appl. Algebra.
[5] , On pseudovarieties, varieties of languages, filters of congruences,
pseudoidentities and related topics, to appear in Algebra Universalis.
[6] J. Almeida e A. Azevedo, Implicit operations on certain classes of semigroups, Semi-
groups and their Applications by S. Goberstein and P.Higgins (Eds.), Proceedings
1986 Chico Conf., D. Reidel, 1987 (1-11).
[7] , The join of the pseudovarieties of R-trivial and L-trivial monoids,
J. Pure Appl. Algebra 60 (1989) 129-137.
[8] , On regular implicit operations, in preparation.
[9] A. Azevedo, The join of the pseudovariety J with permutative pseudovarieties, to
appear in Proceedings Conferência International de Álgebra, Lisboa, 1988.
[10] , Operações Implícitas sobre Pseudovariedades de Semigrupos, Aplicações,
Doctoral dissertation, University of Porto, 1989.
[11] S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York,
1976.
[12] F. W. Lawvere, Functional semantics of algebraic theories, Doctoral dissertation,
University of Columbia, New York, 1963.
[13] J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982)
1-10.


English

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Implicit operations on DS

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