On separability properties in direct products of semigroups
aa r X i v : . [ m a t h . R A ] F e b ON SEPARABILITY PROPERTIES IN DIRECT PRODUCTSOF SEMIGROUPS
GERARD O’REILLY, MARTYN QUICK, NIK RUˇSKUC
Abstract.
We investigate four finiteness conditions related to resid-ual finiteness: complete separability, strong subsemigroup separability,weak subsemigroup separability and monogenic subsemigroup separa-bility. For each of these properties we examine under which conditionsthe property is preserved under direct products. We also consider ifany of the properties are inherited by the factors in a direct product.We give necessary and sufficient conditions for finite semigroups to pre-serve the properties of strong subsemigroup separability and monogenicsubsemigroup separability in a direct product. Introduction, Preliminaries and Summary of Results
The purpose of this paper is to investigate the relationship between directproducts of semigroups and their factors with respect to four separabilityproperties. We begin by defining separability properties in general and thenspecialize to the four of interest within this paper.
Definition 1.1.
For a semigroup S and a collection C of subsets of S , we saythat S has the separability property with respect to C if for any s ∈ S and any C -subset Y ⊆ S \ { s } there exists a finite semigroup P and homomorphism φ : S → P such that φ ( s ) / ∈ φ ( Y ). In this case we say that s can be separated from Y and that φ separates s from Y . Equivalently, S has the separabilityproperty with respect to C if for any s ∈ S and any C -subset Y ⊆ S \ { s } there exists a finite index congruence ρ on S such that [ s ] ρ = [ y ] ρ in S/ρ forall y ∈ Y . In this case we say that ρ separates s from Y .For a semigroup S we say that: • S is monogenic subsemigroup separable (MSS) if S has the sepa-rability property with respect to the collection of all monogenic-subsemigroups; • S is weakly subsemigroup separable (WSS) if S has the separabil-ity property with respect to the collection of all finitely generatedsubsemigroups; • S is strongly subsemigroup separable (SSS) if S has the separabilityproperty with respect to the collection of all subsemigroups; Mathematics Subject Classification.
Primary 20M10; Secondary 20E26, 08A30.
Key words and phrases.
Separability, residual properties, subsemigroup separability,residual finiteness, complete separability, semigroup, direct products, congruence, finiteindex congruence.
Group Property P CS SSS WSS MSS G , H are P ✓ ✓ ✗ ✓ = ⇒ G × H is P [10, Lem 2.4] [9] [2, “The Example”] [12, Thm 4] Table 1.
Separability properties of direct products of groups • S is completely separable (CS) if S has the separability property withrespect to the collection of all subsets.We note that separability properties are finiteness conditions. That is, if P is a separability property then any finite semigroup S has P . This is becausethe identity map will separate any point s from any subset of S \ { s } .The best known separability property is residual finiteness , which can beviewed as the separability property with respect to the collection of singletonsubsets. This is equivalent to having the separability property with respectto the collection of all finite subsets. It was shown in [6, Theorem 2] that thedirect product of two semigroups is residually finite if and only the factorsare residually finite. The fact that residual finiteness of the factors impliesthe residual finiteness of the direct product is universally true for algebraicstructures. However, it is non-trivial to show that if a direct product ofsemigroups is residually finite then both factors are residually finite. Our aimis to determine for which of our four separability properties we can obtainan analogous results. In the cases where we fail to obtain such a result,we investigate further to determine what can be said about the separabilityproperties of direct products.By replacing the word “semigroup” with “group” and replacing “subsemi-group” with “subgroup”, we can define analogous separability properties inthe class of groups. The properties have been widely studied in groups,under various different names. Monogenic subgroup separable groups arealso known as cyclic subgroup separable groups or Π C groups. In [12, The-orem 4], Stebe showed that the direct product of two monogenic-subgroupseparable groups is itself monogenic-subgroup separable. Weakly subgroupseparable groups are also known simply as subgroup separable groups or as locally extended residually finite (LERF) groups. It is not true that the di-rect product of two weakly subgroup separable groups is necessarily weaklysubgroup separable. An example is given by Allenby and Gregorac in [2,“The Example”].
Strongly subgroup separable groups are also known as ex-tended residually finite (ERF) groups. The direct product of two stronglysubgroup separable groups is known to be strongly subgroup separable. In[2] Allenby and Gregorac attribute this result to Mal’cev in [9]. It was shownin [10, Lemma 2.4] that a group is completely separable if and only if it isfinite. Hence, trivially, the direct product of two CS groups is itself CS. Theresults for groups are recorded in Table 1.For groups G and H , both G and H are always isomorphic to subgroups of G × H . As subgroups inherit all four of these properties, an easy extensionof [10, Proposition 2.3], if G × H has one of these properties then so will N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 3 both G and H . However, the situation for algebras in general, and forsemigroups in particular, may not always be so straightforward. Indeed, in[13] de Witt was able to show that there exist monounary algebras A and B such that A is not residually finite but A × B is completely separable.So factors of a direct product of monounary algebras which has separabilityproperty P need not be P themselves. We shall observe that within theclass of semigroups a similar situation occurs with the properties of weaksubsemigroup separability and monogenic subsemigroup separability.Before we begin, we give a taste of how separability properties can interactwith the semigroup direct product by considering the situation where oneof the factors is N . Theorem 1.2.
For a semigroup S , the following fully characterise the sep-arability properties of N × S :(1) N × S is completely separable if and only if S is completely separable;(2) N × S is weakly subsemigroup separable but not strongly subsemigroupseparable if and only if S is residually finite but not completely sep-arable;(3) N × S is not residually finite if and only if S is not residually finite.Proof. (1) As N is CS ([4, Corollary 4]), the result follows from Theorem 2 . ⇒ ) As WSS semigroups are residually finite ([10, Proposition 2.1]), itfollows that S is residually finite by [6, Theorem 2]. The fact S cannot beCS is a consequence of Theorem 2 . ⇐ ) The fact that N × S is WSS follows from [10, Proposition 5.1], whichstates that a residually finite semigroup which has N as homomorphic imageis WSS. The fact that S cannot be SSS follows from Lemma 3 .
1, whichestablishes a more general result.(3) This follows from [6, Theorem 2]. (cid:3)
Remark 1.3.
In the statement of Theorem 1 .
2, the semigroup N can bereplaced by any CS semigroup which has N as a homomorphic image andthe result still holds. Such semigroups include all free semigroups and freecommutative semigroups.The characterisation given in Theorem 1 . N with a SSSsemigroup and the resulting semigroup need not be SSS, even though bothfactors are (and in fact N is CS). On the other hand we can take a semigroupwhich is not WSS but its direct product with N is WSS. Consequently thereis no guarantee that the direct product preserves separability properties northat the factors of a direct product inherit separability properties. GERARD O’REILLY, MARTYN QUICK, NIK RUˇSKUC
Semigroup Property P CS SSS WSS MSS S , T are P ✓ ✗ ✗ ✗ = ⇒ S × T is P Thm 2.3 Ex 3.2 Ex 4.4 Ex 5.12 S × T is P ✓ ✓ ✗ ✗ = ⇒ S, T are P Thm 2.3 Thm 3.3 Ex 4.1 Ex 4.1
Table 2.
Separability properties of direct products of semigroupsBy working through the separability properties in turn, we are able to saywhich of them are preserved under direct products. We also consider whichseparability properties are inherited from the direct product by its factors.These results are recorded in Table 2.It turns out that the properties of strong subsemigroup separability, weaksubsemigroup separability and monogenic subsemigroup separability are notnecessarily preserved in the direct product. This motivates the followingdefinition.
Definition 1.4.
Let P be one of the following properties: strong subsemi-group separability, weak subsemigroup separability or monogenic subsemi-group separability. We say that a semigroup T is P -preserving (in directproducts) if for every semigroup S which has property P , the direct product S × T also has property P .We note that if a semigroup T is P -preserving then T must have property P . This is because the trivial semigroup has property P and T is isomorphicto the direct product of itself with the trivial semigroup. In this paper wefocus on characterising when finite semigroups are P -preserving.We now recall some notions of semigroups and direct products that we shalluse throughout the paper. For a congruence ρ on a semigroup S , we denotethe ρ -class of s ∈ S by [ s ] ρ . For an ideal I ≤ S , we can define a congruence ρ I on S by: s ρ I t if and only if s, t ∈ I or s = t. This congruence is known as the
Rees congruence on S by I . We denotethe quotient of this congruence by S/I and the congruence class of s ∈ S by[ s ] I . Note that if s ∈ S \ I , then [ s ] I = { s } .Green’s relation J on a semigroup is an equivalence relation on S givenby s J t if and only if S sS = S tS , where S is the semigroup S with an identity adjoined. That is, two elementsare J -related if and only if they generate the same two-sided ideal. Green’srelation L on a semigroup is an equivalence relation on S given by s L t if and only if S s = S t. That is, two elements are L -related if and only if they generate the sameleft ideal. Similarly, Green’s relation R on a semigroup is an equivalence N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 5 relation on S given by s R t if and only if sS = tS . That is, two elements are R -related if and only if they generate the sameright ideal. Green’s relation H on a semigroup is an equivalence relation on S given by s H t if and only if s L t and s R t. An H -class H is a subgroup of S if and only if H = H .For semigroups S and T , the projection map π S : S × T → S is given by( s, t ) s . Similarly, π T : S × T → T is given by ( s, t ) t . Projection mapsare homomorphisms. Throughout N denotes the set { , , , . . . } .This paper is organised by property. In Section 2 we show that the di-rect product of two semigroups is CS if and only if both semigroups areCS (Theorem 2 . . . . Complete Separability
Complete separability proves to be the best behaved of our separabilityproperties with respect to the direct product. In order to show this, we willuse a characterisation of complete separability given by Golubov in [4]. Thischaracterisation depends upon the following definition.
Definition 2.1.
For a semigroup S and a, b ∈ S define[ a : b ] = { ( u, v ) ∈ S × S | ubv = a } . Using this definition, we have the following characterisation of CS semi-groups.
Theorem 2.2. [4, Theorem 1]
A semigroup S is completely separable if andonly if for each a ∈ S the set { [ a : s ] | s ∈ S } is finite. Using this, we give the following result.
Theorem 2.3.
The direct product of semigroups S and T is completelyseparable if and only if S and T are completely separable. GERARD O’REILLY, MARTYN QUICK, NIK RUˇSKUC
Proof. ( ⇐ ) The fact that the direct product of two CS semigroups is CSwas shown by Golubov in [5, Lemma 2]. Here we present an independentproof. Assume that S and T are CS. Let ( s, t ) ∈ S × T . There exists a finitesemigroup U and homomorphism φ : S → U such that φ ( s ) / ∈ φ ( S \ { s } ).There exists a finite semigroup V and homomorphism ψ : T → V such that ψ ( t ) / ∈ ψ ( T \{ t } ). Then φ × ψ : S × T → U × V given by ( φ × ψ )( a, b ) =( φ ( a ) , ψ ( b )) is a homomorphism that separates ( s, t ) from its complement.( ⇒ ) We will prove the contrapositive, so assume that S is not CS. We showthat S × T is not CS. By Theorem 2 . a ∈ S such thatthe set { [ a : s ] | s ∈ S } is infinite. Let ∼ be a finite index congruence on S × T . Fix t ∈ T . Then there exist b, c ∈ S such that [ a : b ] = [ a : c ] and( b, t ) ∼ ( c, t ). Assume, without loss of generality, that there exist u, v ∈ S such that ubv = a but ucv = a . We split into two cases. The first case isthat u, v ∈ S . Then( a, t ) = ( u, t )( b, t )( v, t ) ∼ ( u, t )( c, t )( v, t ) = ( ucv, t ) . The second case is that either u = 1 or v = 1. We will deal with the casethat u = 1. A similar argument deals with the case that v is not in S . Then( a, t ) = ( b, t )( v, t ) ∼ ( c, t )( v, t ) = ( cv, t ) = ( ucv, t ) . In either case we cannot separate ( a, t ) from its complement. Therefore S × T is not CS. (cid:3) Strong Subsemigroup Separability
Strong subsemigroup separability is not as well behaved as complete sep-arability or residual finiteness with respect to the direct product. Indeed,the direct product of two SSS semigroups need not be itself SSS. Golubovshowed this in [5] with the direct product of two infinite SSS semigroups.We show that even when one of the factors is finite, the direct product oftwo SSS semigroups need not be SSS (Example 3 . . T = h x i is either isomorphic to N or it is finite. In the case that T is finite, there exist positive integers m and r such that x m = x m + r . Thesmallest such value of m is called the index of T and the smallest such valueof r is called the period of T . For more on monogenic semigroups see [8, The-orem 1.2.2]. To simplify statements, we will say that the infinite monogenicsemigroup has index ∞ . Lemma 3.1.
Let S be a semigroup which is not completely separable and let T be a monogenic semigroup with index m ≥ . Then S × T is not stronglysubsemigroup separable.Proof. Let x be the generator of T . As S is not CS, by Theorem 2 . a ∈ S such that the set { [ a : s ] | s ∈ S } is infinite. Let ∼ be afinite index congruence on S × T . Then there exist b, c ∈ S such that [ a : b ] = N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 7 [ a : c ] and ( b, x ) ∼ ( c, x ). Assume, without loss of generality, that there exist u, v ∈ S such that ubv = a but ucv = a . Let U = h ( S \{ a } ) ×{ x }i ≤ S × T .Since m ≥ x / ∈ { x k | k > } , and hence ( a, x ) U .We split into two cases. The first is that u, v ∈ S . Then( a, x ) = ( u, x )( b, x )( v, x ) ∼ ( u, x )( c, x )( v, x ) = ( ucv, x ) ∈ U. The second case is when either u = 1 or v = 1. We will deal with the casethat u = 1. Then( a, x ) = ( b, x )( v, x ) ∼ ( c, x )( v, x ) = ( ucv, x ) ∈ U. In either case we cannot separate ( a, x ) from U . A similar argument dealswith the case that v is not in S . Therefore S × T is not SSS. (cid:3) As there exist SSS semigroups which are not CS, for example [10, Exam-ples 5.4 and 5.11], we can use Lemma 3 . Example 3.2.
As an immediate consequence of Lemma 3 .
1, if we takethe direct product of an SSS semigroup S which is not CS with the finite4-nilpotent semigroup h x | x = x i , the resulting semigroup is not SSS.Despite the negative nature of Lemma 3 .
1, it actually plays a key part inthe proof of the following result.
Theorem 3.3. If S × T is strongly subsemigroup separable then both S and T are strongly subsemigroup separable.Proof. Assume S × T is SSS. Without loss of generality we show that S isSSS. We split into two cases.The first case is that T contains an idempotent. Then S is isomorphic to asubsemigroup of the SSS semigroup S × T and hence S must also be SSS by[10, Proposition 2.3].The second case is that T has no idempotents. For a contradiction assumethat S is not SSS. Then S is certainly not CS. As T is idempotent free,it contains a copy of N . Then S × N ≤ S × T . But S × N is not SSS byLemma 3 .
1. This contradicts the strong subsemigroup separability of S × T and therefore it must be the case that S is SSS. (cid:3) The rest of this section is dedicated to establishing when a finite semigroup isSSS-preserving. To do this we show that SSS semigroups enjoy a separabilityproperty, in the sense of Definition 1 .
1, with respect to a more general classof subsets than just subsemigroups.
Proposition 3.4.
Let S be a strongly subsemigroup separable semigroupand let V ⊆ S such that V n +1 ⊆ V for some n ∈ N and let s ∈ S \ V . Then s can be separated from V . GERARD O’REILLY, MARTYN QUICK, NIK RUˇSKUC
Proof.
We proceed by induction on n . The base case n = 1 corresponds to V being a subsemigroup, and the result follows from S being SSS.Now consider n >
1. Our inductive hypothesis is that for all U ⊆ S suchthat U k +1 ⊆ U for some k ∈ { , , . . . , n − } , and for all s ′ ∈ S \ U , we canseparate s ′ from U . Let V and s be as in the statement of the proposition.If s / ∈ h V i then as S is SSS, we can separate s from h V i and in particularfrom V .So suppose that s ∈ h V i . Note that as V n +1 ⊆ V , we have that h V i = V ∪ V ∪ · · · ∪ V n and that V n is a subsemigroup of S . Let L = { ℓ ∈ V | s ∈ ℓV i , ≤ i ≤ n − } , which is some set of left-divisors of s . As s ∈ h V i , we have that L is non-empty. Let Z = V \ L . Then we claimthat s / ∈ h Z i . Indeed if s ∈ h Z i , then s = zw for some z ∈ Z and w ∈ Z k ,where k ∈ N ∪ { } . If k = 0 then s = z ∈ Z ⊆ V , which is a contradiction.Otherwise, as V n +1 ⊆ V , we have w ∈ V i for some i ∈ { , , . . . , n } . If w ∈ V n , then s ∈ V n +1 ⊆ V , which is a contradiction. Then w ∈ V i forsome 1 ≤ i ≤ n −
1. But then z ∈ L , which is a contradiction. So s / ∈ h Z i . As S is SSS, s can be separated from h Z i and in particular s can be separatedfrom Z .For i ∈ { , , . . . , n − } , define X i = V ∩ V i +1 . Then we claim that X n − i +1 i ⊆ X i . (1)To see this let x , x , . . . , x n − i +1 ∈ X i . Firstly, as X i = V ∩ V i +1 , we have x ∈ V i +1 and x , . . . , x n − i +1 ∈ V . Then x x . . . x n − i +1 ∈ V ( i +1)+( n − i +1 − = V n +1 ⊆ V. Secondly, noting that n − i + 1 ≥
2, we have x , x ∈ V i +1 . Then x x . . . x n − i +1 ∈ V i +1)+( n − i +1 − = V n + i +1 ⊆ V i +1 . Hence x x · · · x n − i +1 ∈ V ∩ V i +1 = X i and we conclude X n − i +1 i ⊆ X i and(1) holds. As s / ∈ V we have that s / ∈ X i .Since 1 ≤ n − i ≤ n − X n − i +1 i ⊆ X i , by our inductive hypothesis wecan separate s from each X i . For i ∈ { , . . . , n − } , define L i = { ℓ ∈ L | s ∈ ℓV i } . Note that L = S ≤ i ≤ n − L i . We show that s can be separatedfrom each L i . Suppose that s cannot be separated from some L i . We claimthat sV n − ⊆ V i ∩ V n . First for a contradiction, assume that there exists u ∈ V n − such that su ∈ S \ V n . Then as V n is a subsemigroup and S isSSS it must be the case that su can be separated from V n . Let ∼ be a finiteindex congruence which separates su from V n . As s cannot be separatedfrom V , there exists v ∈ V such that s ∼ v . But then su ∼ vu ∈ V n which is a contradiction. Hence if s cannot be separated from L i , we have sV n − ⊆ V n . As L i is non-empty, we have that s ∈ V i +1 . But as s ∈ V i +1 and V n +1 ⊆ V , we have that sV n − ⊆ V i . Hence sV n − ⊆ V n ∩ V i asclaimed. From this we get sV n ⊆ V ∩ V i +1 = X i . (2)Now let ∼ be a finite index congruence that separates s from X i . As s cannot be separated from L i , there exists ℓ ∈ L i such that ℓ ∼ s . But as N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 9 ℓ ∈ L i , there exists u ∈ V i such that ℓu = s . Then, by the compatibility of ∼ we have s = ℓu ∼ su ∼ su ∼ · · · ∼ su n − ∼ su n . By the transitivity of ∼ we obtain that s ∼ su n . As u ∈ V i , we have that u n ∈ V in . But as V n +1 ⊆ V , we conclude that u n ∈ V n . So by Equation(2) we have su n ∈ sV n ⊆ X i . That is ∼ does not separate s from X i . Thisis a contradiction and so s can be separated from L i .As V = Z ∪ L ∪ L ∪ . . . L n − , and for each of the sets in this finite unionthere exists a finite index congruence which separates s from said set, theintersection of these congruences is a finite index congruence which separates s from V . (cid:3) The fact that SSS semigroups satisfy the additional separability property ofProposition 3 . S and an element s ∈ S , we say that s is decomposable if s ∈ S . In this case, there exist t, u ∈ S such that s = tu .Otherwise we say that s is indecomposable. Theorem 3.5.
A finite semigroup P is strong subsemigroup separabilitypreserving if and only if every element of P is indecomposable or belongs toa subgroup.Proof. ( ⇒ ) We prove the contrapositive so assume that p ∈ P is not con-tained in a subgroup but there exist s, t ∈ P such that st = p . As p is notcontained in a subgroup, we have that p n = p for all n ≥
2. Let G be aninfinite SSS group (e.g. [10, Example 5.4]). Let U ≤ G × P be generatedby the set { ( g, p ) | g ∈ G \ { G }} . As p n = p for all n ≥
2, we have that(1 G , p ) / ∈ U . Let ∼ be a finite index congruence on G × P . Then there exist g, h ∈ G with g = h such that ( g, s ) ∼ ( h, s ). Then(1 G , p ) = ( g, s )( g − , t ) ∼ ( h, s )( g − , t ) = ( hg − , p ) ∈ U. Hence G × P is not SSS and therefore P is not SSS-preserving.( ⇐ ) Now assume that P is a finite semigroup in which every element notcontained in a subgroup is indecomposable. Let S be an SSS semigroup,let U ≤ S × P and let ( s, p ) ∈ ( S × P ) \ U . If s / ∈ π S ( U ) ≤ S , then wecan separate ( s, p ) from U by factoring through S and invoking the strongsubsemigroup separability of S . If p / ∈ π P ( U ), then π P separates ( s, p ) from U .Now assume that both s ∈ π S ( U ) and p ∈ π P ( U ). If p is not contained withina subgroup of P , then p is indecomposable. Then ( s, p ) is indecomposablein S × P . Let I = ( S × U ) \ { ( s, p ) } . Then I is an ideal of finite complementin S × U and [( s, p )] I = { ( s, p ) } . In particular, ( s, p ) is separated from U inthe Rees quotient of S × P by I .The final case to consider is that p is contained in a subgroup of P . Thenfor some n ∈ N , we have that p n +1 = p . Let V = π S ( U ∩ ( S × { p } ). Firstnote that s / ∈ V as ( s, p ) / ∈ U . Secondly, as p n +1 = p , we have V n +1 ⊆ V . Then by Proposition 3 . s can be separated from V . So thereexists a finite semigroup Q and a homomorphism φ : S → Q such that φ ( v ) / ∈ φ ( V ). Define φ : S × P → Q × P by ( a, b ) ( φ ( a ) , b ). Then φ is ahomomorphism which separates ( s, p ) from U . Hence S × P is SSS and so P is SSS-preserving. (cid:3) Remark 3.6.
As the set S is an ideal of a semigroup S , Theorem 3 . P is SSS-preserving if and onlyif P is the ideal extension of a union of groups by a null semigroup. Corollary 3.7.
The following families of finite semigroups are strong sub-semigroup separability preserving: groups, Clifford semigroups, completelysimple semigroups, completely regular semigroups, bands, and null semi-groups.
Corollary 3.8.
A finite monoid is strong subsemigroup separability preserv-ing if and only if it is a union of groups.
We conclude this section with some open problems.
Open Problem 3.9.
Is there a characterisation of when the direct productof two strongly subsemigroup separable semigroups is itself strongly sub-semigroup separable?
Open Problem 3.10.
Is it true that the direct product of two stronglysubsemigroup separable semigroups is weakly subsemigroup separable?4.
Weak Subsemigroup Separability
The behaviour of weak subsemigroup separability with respect to the di-rect product is more complicated than that of the separability propertieswe have already discussed. This is exhibited in Open Problem 3 .
10, whichstates that even when we strengthen the factor semigroups to be SSS, westill do not know if this guarantees that the direct product is WSS. Wealso see for the first time an example of a separability property which isnot necessarily inherited by the factors of a direct product (Example 4 . . . . Example 4.1.
Consider the semigroup N × Z . As this is a residually finitesemigroup with N as a homomorphic image, it is WSS by [10, Proposition5.1]. However, Z is not WSS by [10, Example 2.6]. N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 11
Remark 4.2.
While [10, Example 2.6] states that Z is not WSS, the sub-semigroup used in demonstrating this is N . As N is monogenic, we have that Z is not MSS. Therefore N × Z is also an example of a direct product whichis MSS but one of the factors is not MSS.It also turns out that the direct product of two WSS semigroups need notbe WSS, even when one of the factors in finite. To show this we first needto construct a specific instance of a WSS semigroup. Definition 4.3.
Let FC be the free commutative semigroup on the set { a, b } . Let N = { x z | z ∈ Z } ∪ { } be a null semigroup with zero element 0.Let φ : FC → Z be given by a i b j i − j . Consider the set S [FC , Z , φ ] =FC ∪ N . We extend the multiplication on FC and N in the followingmanner. For w ∈ FC and x z ∈ N , define x z · w = x z + φ ( w ) ,w · x z = x z − φ ( w ) , · w = w · . Then S [FC , Z , φ ] is an example of semigroup of type given in [10, Construc-tion 5.5] and is WSS by [10, Proposition 5.8]. Example 4.4.
Let S = S [FC , Z , φ ] and let L be a non-trivial left-zerosemigroup. Then S × L is not WSS.Let y, z ∈ L be distinct. Let U = h ( a, y ) , ( x , z ) i ≤ S × L. Then one computes that U = { ( a i , y ) | i ∈ N } ∪ { ( x i , z ) | i ∈ N } ∪ { ( x i , y ) | i ∈ Z } ∪ { (0 , y ) , (0 , z ) } . In particular, ( x , z ) / ∈ U . Let ∼ be a finite index congruence on S × L .Then there exist i, j ∈ N with i < j such that ( x i , z ) ∼ ( x j , z ). Then( x , z ) = ( x i , z )( b i , z ) ∼ ( x j , z )( b i , z ) = ( x j − i , z ) ∈ U. Hence S × L is not weakly subsemigroup separable. By a similar argument, itcan be shown that the direct product of S = S [FC , Z , φ ] with a non-trivialright-zero semigroup is not WSS.Example 4 . . .
10. Adding yet another twist tothe story, the following theorem shows that finite nilpotent semigroups areWSS-preserving, even though this class contains semigroups which are nei-ther SSS-preserving nor MSS-preserving. For example the semigroup withpresentation h x | x = x i is a finite 3-nilpotent semigroup which fails tomeet the criteria to be either SSS-preserving or MSS-preserving. Theorem 4.5.
The direct product of a weakly subsemigroup separable semi-group with a residually finite nilpotent semigroup is weakly subsemigroupseparable.Proof.
Let S be a WSS semigroup and let N be a residually finite k -nilpotentsemigroup with zero element 0, for some k ∈ N . Let U ≤ S × N be finitelygenerated and let ( s, n ) ∈ ( S × N ) \ U . Fix a finite generating set for U ,which has the form( X × { } ) ∪ ( X × { n } ) ∪ · · · ∪ ( X j × { n j } ) , where n , n , . . . , n j ∈ N \ { } and X , X , . . . , X j are finite subsets of S .Let X = S ji =0 X i and let T = h X i ≤ S .Let Z = π N ( U ) = { z , z , z , · · · , z m } where z = 0. Note Z is finite as k -nilpotent semigroups are locally finite. For 0 ≤ i ≤ m let Y i = π S ( U ∩ ( S × { z i } )) . Then for 1 ≤ i ≤ m the set Y i is finite. To see this first note that Y i isa subset of T . Then for y ∈ Y i , we can write y as a product of elementsof X . The maximum length of the product is k −
1, as N is a k -nilpotentsemigroup and z i is a non-zero elements of N . As X is a finite set, there areonly finitely many such y and so Y i is finite.Now consider Y . Certainly T k ⊆ Y as N is a k -nilpotent semigroup. Wehave that Y \ T k is finite as any element of Y \ T k can be expressed asproduct over X of length at most k − s / ∈ π S ( U ) = T then we can separate ( s, n ) from U by factoring through S and invoking the weak subsemigroup separability of S . Similarly, if t / ∈ π N ( U ) = Z then we can separate ( s, n ) from U by factoring through N andusing the residual finiteness of N .Now assume that s ∈ π S ( U ) and n ∈ π N ( U ). Then either(i) n = 0 and s / ∈ Y , or(ii) n = z i for some 1 ≤ i ≤ m and s / ∈ Y i .(i) First note that T k ≤ S is a finitely generated by the set X k ∪ X k +1 ∪ · · · ∪ X k − . As s / ∈ T k and S is WSS, there exists a finite semigroup P and homo-morphism φ : S → P such that φ ( s ) / ∈ φ ( T k ). As s / ∈ Y \ T k and S is residually finite, there exists a finite semigroup P and homomorphism φ : S → P such that φ ( s ) / ∈ φ ( Y \ T k ). Therefore the homomorphism φ : S → P × P given by a ( φ ( a ) , φ ( a )) separates s from Y .As N is residually finite, there exists a finite semigroup Q and homomor-phism ψ : N → Q such that ψ (0) / ∈ ψ ( Z \ { } ). Then the homomorphism φ × ψ : S × N → ( P × P ) × Q given by ( a, b ) ( φ ( a ) , ψ ( b )) separates ( s, n )from U .(ii) As S is WSS it is residually finite by [10, Proposition 2.1] there exists afinite semigroup P and homomorphism φ : S → P such that φ ( s ) / ∈ φ ( Y i ). N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 13 As N is residually finite, there exists a finite semigroup Q and homomor-phism ψ : N → Q such that ψ ( z i ) / ∈ ψ ( Z \{ z i } ). Then φ × ψ : S × N → P × Q given by ( a, b ) ( φ ( a ) , ψ ( b )) separates ( s, n ) from U . This completes theproof that S × N is WSS. (cid:3) A characterisation for when a finite semigroup is weakly subsemigroup sep-arability preserving is not known. We leave this as one of the open problemsconcerning weak subsemigroup separability and direct products. But beforewe state the open problems, we show that the direct product of two WSSsemigroups is MSS.
Theorem 4.6.
The direct product of two weakly subsemigroup separablesemigroups is monogenic subsemigroup separable.Proof.
Let S and T be WSS semigroups. Let U = h ( s, t ) i ≤ S × T be amonogenic subsemigroup and let ( x, y ) ∈ ( S × T ) \ U . If x / ∈ π S ( U ) = h s i thenwe can separate ( x, y ) from U by factoring through S and using the weaksubsemigroup separability of S . By a similar argument, we can separate( x, y ) from U if y / ∈ π T ( U ).Now consider the case that x ∈ π S ( U ) and y ∈ π S ( T ). Then x = s i and y = t j , where i, j ∈ N such that i = j , s i = s j and t i = t j . Firstly we considerthe case when at least one of h s i and h t i is infinite. Without loss of generality,assume that h s i ∼ = N . In this instance s i / ∈ h s i +1 , s i +2 , . . . , s i +1 i = { s k | k ≥ i + 1 } . Hence we can separate ( x, y ) from { ( s k , t k ) | k ≥ i + 1 } by factoringthrough S and using the weak subsemigroup separability of S . That is,there exists a finite semigroup P and a homomorphism φ : S × T → P such that φ ( x, y ) = φ ( s n , t n ) for all n ≥ i + 1. As S and T are bothWSS, they are both residually finite by [10, Proposition 2.1]. Hence S × T is residually finite by [6, Theorem 2]. Then we can separate ( x, y ) fromthe finite set { ( s k , t k ) | k ≤ i } . That is, there exists a finite semigroup P and homomorphism φ : S × T → P such that φ ( x, y ) = φ ( s k , t k )for 1 ≤ k ≤ i . Then the homomorphism φ : S × T → P × P given by φ ( a, b ) = ( φ ( a, b ) , φ ( a, b )) separates ( x, y ) from U .The final case to consider is when both h s i and h t i are finite, in which case U is finite. As S × T is residually finite we can separate ( x, y ) from U . Hence S × T is MSS as desired. (cid:3) We conclude this section with some open problems.
Open Problem 4.7.
Is there a characterisation of when a finite semigroupis weakly subsemigroup preserving?
Open Problem 4.8.
Is there a characterisation of when a direct product oftwo weakly subsemigroup separable semigroups is itself weakly subsemigroupseparable?
Open Problem 4.9.
If the direct product of two semigroups is weaklysubsemigroup separable, is at least one of the factors weakly subsemigroupseparable? 5.
Monogenic subsemigroup Separability
We have already seen that if a direct product of two semigroups is MSS,then it need not be that both factors are MSS (Example 4 . . . . A , which we will define shortly. In showing that A is group embeddable we use a criterion given by Adian, which we nowreview. Definition 5.1.
Let h X | R i be a presentation. Consider a relation ( u, w ) ∈ R . The left pair of ( u, w ) is the pair ( x, y ), where x is the leftmost letterof u and y is the leftmost letter of w . The right pair of ( u, w ) is the pair( z, t ), where z is the rightmost letter of u and t is the rightmost letter of w .The left graph of the presentation is the graph with vertex set X such that { x, y } is an edge if and only if ( x, y ) is a left pair of some relation. Note thatmultiple edges and loops are allowed. The right graph of the presentation isthe graph with vertex set X such that { z, t } is an edge if and only if ( z, t ) isa right pair of some relation. The presentation h X | R i is said to have no cycles if both its left graph right graph have no cycles (here loops are cyclesand multiple edges create cycles). Theorem 5.2. ([1, Theorem 2.3])
If a presentation has no cycles then thenatural mapping from the semigroup given by the presentation to the groupgiven by the presentation is an embedding.
Corollary 5.3.
The mapping φ given by a x, b y, c y − x − y, from the semigroup A given by the presentation h a, b, c | ab c = b i to thefree group FG on the set { x, y } is an embedding.Proof. By Theorem 5 .
2, the natural map from A to the group G given bythe group presentation h x, y, z | xy z = y i is an embedding. It can beeasily seen that this group is free, by use of a single Tietze transformationeliminating the generator z . (cid:3) Definition 5.4.
In the semigroup A with presentation h a, b, c | ab c = b i ,the strings abbc and b represent the same element. Therefore we can define N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 15 a rewriting system on A that replaces the string abbc by b . As the singlerewriting rule is length reducing, the rewriting system is terminating. Also,as ab c does not overlap with itself, the rewriting system is locally confluentand hence confluent. For more on rewriting systems see [3, Section 1.1].Therefore, each element of A is represented by a unique word in { a, b, c } + ,where this representative does not contain abbc as a subword. Such a wordis said to be in normal form . Note that a contiguous subword of a normalform word is also a word in normal form. We shall therefore consider theunderlying set of A to be the set of all words over { a, b, c } in normal form.Multiplication is concatenation, except in the case where concatenation cre-ates instances of abbc as subwords, in which case the rewriting rule is appliedto convert the product into normal form. Lemma 5.5.
Let A , FG and φ : A → FG be as in Corollary . . Then φ ( w ) = ǫ for all w ∈ A .Proof. Let w be an element of A . We proceed by a case analysis based uponthe number of occurrences of contiguous strings of the letter c appearing inthe word w . Case 1.
The first case in when there are no occurrences of the letter c in w . Then w ∈ { a, b } + . As φ rewrites an occurrence of a with an x and anoccurrence of b with a y , we have that φ ( w ) is a non-empty word, as desired. Case 2.
Now we consider the case when w contains precisely one string ofthe letter c . We split into three subcases. Case 2a.
Consider w = c n , where n ≥
1. Observe that φ ( c n ) = y − ( x − y − ) n − x − y. Hence the assertion of the lemma holds. For future cases note that φ ( c n )ends with the suffix x − y . Case 2b.
Consider w = uc n , where u ∈ { a, b } + and n ≥
1. We claimthat when concatenating the words φ ( u ) and φ ( c n ), there are at most twocancelling pairs of letters. Indeed, if there were three cancelling pairs ofletters, then φ ( u ) would end with xy . This follows since by Case (2a) weknow that φ ( c n ) begins with y − x − . Hence u would end with ab . But inthis case uc n contains the string ab c , contradicting that uc n is in normalform. It follows that ( x − y − ) n − x − y is a suffix of φ ( uc n ) and hence φ ( uc n )is non-empty. Again we note that φ ( uc n ) ends with x − y . For future caseswe consider when φ ( uc n ) can begin with a negative letter. In such a case, theentirety of φ ( u ) has been cancelled by φ ( c n ). By our analysis of cancellationbetween φ ( u ) and φ ( c ) there are only two options: u = b or u = b . Case 2c.
Consider w = uc n v , where u ∈ { a, b } ∗ , v ∈ { a, b } + and n ≥ φ ( v ) consists of positive letters. By Cases (2a) and (2b), φ ( uc n ) endswith a positive letter. Then when concatenating φ ( uc n ) and φ ( v ), there canbe no pairs of cancelling letters. Hence φ ( uc n v ) is non-empty as desired. Case 3.
Finally we consider the case when w contains more than one stringof the letter c . We can decompose w = w w . . . w k v where: • w = uc n , where u ∈ { a, b } ∗ and n ≥ • w i = u i c n i , where u i ∈ { a, b } + and n i ≥ ≤ i ≤ k ; • v ∈ { a, b } ∗ ; and k ≥ i we have that φ ( w i ) is a non-emptyword. We now claim that for 1 ≤ i ≤ k , when we concatenate φ ( w i ) and φ ( w i +1 ) there is at most one cancelling pair of letters. We have already ob-served in Cases (2a) and (2b) that φ ( w i ) must end with x − y . Therefore forcancellation to occur, φ ( w i +1 ) must begin with y − . By the final observationof Case (2b), φ ( w i +1 ) can only begin with a negative letter if u i +1 = b or u i +1 = b . In the first case, φ ( w i +1 ) = y − ( x − y − ) n i +1 − x − y and thereis precisely one cancelling pair when we concatenate φ ( w i ) and φ ( w i +1 ). Inthe second case, φ ( w i +1 ) = ( x − y − ) n i +1 − x − y and no cancellation occurswhen we concatenate φ ( w i ) and φ ( w i +1 ) completing the proof of the claim.Note that if cancellation occurs, than the cancelling pair is yy − .Now consider φ ( w ) φ ( w ) . . . φ ( w k ) φ ( v ). As already observed, φ ( w ) and φ ( w ) both end with x − y . By the claim of the previous paragraph, thereis at most one cancelling pair of letters, yy − , between φ ( w ) and φ ( w ).So it must be the case that φ ( w w ) also ends with x − y . Continuing inthis manner, we conclude that φ ( w w . . . w k ) ends with x − y . As φ ( v ) iseither empty or consists of positive letters, there is no cancellation between φ ( w w . . . w k ) and φ ( v ) and we conclude φ ( w ) is non-empty, completing theproof of this case and of the lemma. (cid:3) Before we proceed to show that A is MSS but not WSS, we introduce thenotion of stability. This is needed in showing that A is not WSS. Definition 5.6.
A semigroup S is stable if both a J ab = ⇒ a R ab and a J ba = ⇒ a L ba. Lemma 5.7. [11, Theorem A.2.4]
Finite semigroup are stable.
Remark 5.8.
In a stable semigroup if x J x , then we have that x R x and x L x . In other words x H x . Then H ∩ H = ∅ , where H is the H -class of x , and we conclude that H is a group.We are now ready to establish the separability properties of the semigroup A . Proposition 5.9.
The semigroup A given by the presentation h a, b, c | ab c = b i is monogenic subsemigroup separable but not weakly subsemigroupseparable.Proof. Let T = h u i ≤ A be a monogenic subsemigroup and let v ∈ A \ T .Let FG and φ : A → FG be as in Corollary 5 .
3. As φ is an embedding, wehave that φ ( v ) / ∈ φ ( T ). Let H ≤ FG be the cyclic subgroup with generator φ ( u ). First we show that φ ( v ) / ∈ H . N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 17
For a contradiction, assume that φ ( v ) ∈ H . As φ ( v ) / ∈ φ ( T ), we havethat φ ( v ) / ∈ { φ ( u ) n | n ∈ N } . By Lemma 5 . φ ( v ) = ǫ and therefore φ ( v ) = φ ( u ) . Therefore φ ( v ) = φ ( u ) − n for some n ∈ N . But then φ ( vu n ) = φ ( v ) φ ( u n ) = ǫ . This contradicts Lemma 5 . φ ( v ) / ∈ H .As FG is weakly subgroup separable [7, Theorem 5.1], there exists a finitegroup G and homomorphism ψ : FG → G such that ψ ( φ ( v )) / ∈ ψ ( H ). Inparticular, ψ ◦ φ : A → G is a homomorphism from A to a finite semigroupsuch that ( ψ ◦ φ )( u ) / ∈ ( ψ ◦ φ )( T ). Hence A is MSS.To show that A = h a, b, c | ab c = b i is not weakly subsemigroup separable,consider the subsemigroup V = h b , b i . As h b i ∼ = N , we have b ∈ A \ V . Let P be a finite semigroup and let σ : A → P be a homomorphism. The relation ab c = b ensures that b J b and hence σ ( b ) J σ ( b ). But as P is finite we havethat H σ ( b ) is a group by Lemma 5 . .
8. Hence σ ( V ) = h σ ( b ) i is a finite cyclic group and in particular σ ( b ) ∈ σ ( V ). Therefore A is notweakly subsemigroup separable. (cid:3) The semigroup A will prove crucial in characterising which finite semigroupsare MSS-preserving. In fact our characterisation will stretch to residuallyfinite periodic semigroups. Note that a periodic semigroup is MSS if and onlyif it is residually finite. Periodic residually finite semigroups are MSS becauseevery monogenic subsemigroup of such a semigroup is finite. The fact thatMSS semigroups are residually finite is a consequence of [10, Proposition 2.1].Although the statement of [10, Proposition 2.1] asserts that WSS semigroupsare residually finite, the proof of this assertion only uses weak subsemigroupseparability to separate elements from monogenic subsemigroups, and thusalso serves as a proof that MSS semigroups are residually finite. Theorem 5.10.
A residually finite periodic semigroup is monogenic sub-semigroup separability preserving if and only if it is a union of groups.Proof. ( ⇐ ) Let T be a residually finite periodic semigroup which is a unionof groups. Let S be a MSS semigroup and let U = h ( s, t ) i ≤ S × T . Let( x, y ) ∈ ( S × T ) \ U . We separate into cases. Note some cases may overlap. Case 1.
Suppose h s i ≤ S is finite. As T is periodic, we also have that h t i is finite. Therefore U is also finite. As S is MSS it is also residually finite([10, Proposition 2.1]) and therefore S × T is residually finite. Hence we canseparate ( x, y ) from U . Case 2.
Suppose that x / ∈ π S ( U ); that is, x / ∈ h s i ≤ S . We can separate( x, y ) from U by factoring through S and invoking the monogenic subsemi-group separability of S . Case 3.
Suppose that y / ∈ π T ( U ). As π T ( U ) = h t i and T is periodic, wehave that π T ( U ) is finite. So we can separate ( x, y ) from U by factoringthough T and invoking the residual finiteness of T . Case 4.
Now suppose that h s i ∼ = N , x ∈ π S ( U ) and y ∈ π T ( U ). Let r ∈ N be minimal such that t r +1 = t . Such an r exists as T is periodic and a union of groups. As x ∈ π S ( U ), we have that x = s i for some i ∈ N . As y ∈ π T ( U ),we have that y = t j for some j ∈ { , , . . . , r } . Observe that π S ( U ∩ ( S × { t j } )) = { s j + rn | n ≥ } . As ( x, y ) / ∈ U , it must be the case that i j (mod r ). We split into twosubcases. Case 4a.
First we will deal with the case j = r . In this case, y is anidempotent and we have π S ( U ∩ ( S × { t r } )) = h s r i . Hence x = s i / ∈ h s r i . As S is MSS, there exists a finite semigroup P andhomomorphism φ : S → P such that φ ( x ) / ∈ φ ( h s r i ). As T is residuallyfinite, there exists a finite semigroup P and homomorphism φ : T → P such that φ ( y ) = φ ( π T ( U ) \ { y } ). Then φ : S × T → P × P given by( a, b ) ( φ ( a ) , φ ( b )) is a homomorphism that separates ( x, y ) from U . Case 4b.
Now assume that j ∈ { , , . . . , r − } . Let k be such that j + k = r . Then as i j (mod r ), we have that i + k j + k (mod r ).Hence ( s i + k , t j + k ) = ( s i + k , t r ) / ∈ U . By Case 4a we can separate ( s i + k , t r )from U .We now show that we can separate ( x, y ) = ( s i , t j ) from U . For a contra-diction suppose it cannot be separated. Let ∼ be a finite index congruencewhich separates ( s i + k , t r ) from U . As ( s i , t j ) cannot be separated from U ,there exists ℓ ∈ N such that ( s i , t j ) ∼ ( s, t ) ℓ . But then( s i + k , t r ) = ( s i , t j )( s, t ) k ∼ ( s, t ) ℓ ( s, t ) k ∈ U. This contradicts that ∼ separates ( s i + k , t r ) from U . Hence ( x, y ) can beseparated from U , completing the proof of this case and establishing that S × T is MSS.( ⇒ ) Now suppose that T is a residually finite periodic semigroup which isnot a union of groups. We will show that T is not MSS-preserving. As T is not a union of groups, there exists an element t ∈ T such that for all n ≥ t n = t . Let m be the index and let r be the period ofthe monogenic subsemigroup h t i . We have that m ≥ t m = t m + r . Let i = m − A be the monogenic subsemigroup separable semigroup givenby the presentation h a, b, c | ab c = b i . Let U = h ( b, t ) i ≤ A × T . Observethat π A ( U ∩ ( A × { t i + r } ) = { b i + nr | n ≥ } . (3)Then ( b i , t i + r ) / ∈ U .Let d ∈ { m, m + 1 , . . . , m + r − } be such that d ≡ r ). Then itmust be the case that t d is an idempotent. Furthermore, t i + r t d = t i + r . Let ρ be a finite index congruence on A × T . Then[( a, t d )] ρ [( b , t d )] ρ [( c, t d )] ρ = [( ab c, t d )] ρ = [( b, t d )] ρ . As [( b, t d )] ρ [( b, t d )] ρ = [( b , t d )] ρ we have that [( b, t d )] ρ J [( b, t d )] ρ . Since( A × T ) /ρ is finite, we conclude that the H -class of [( b, t d )] ρ is a finite N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 19 group by Lemma 5 . .
8. In particular there exists k > b, t d )] kρ = [( b, t d )] ρ . From this we conclude that [( b ℓ , t p )] ρ [( b, t d )] k − ρ =[( b ℓ , t p )] kρ for all ℓ ∈ N and p ∈ { m, m + 1 , . . . , m + r − } . But then[( b i , t i + r )] ρ = [( b i , t i + r )] ρ [( b, t d )] ( k − rρ = [( b i +( k − r , t i + r )] ρ . From Equation (3), we have that ( b i +( k − r , t i + r ) ∈ U . So ρ does not sep-arate ( b i , t i + r ) from U . As ρ was arbitrary we conclude that A × T is notMSS and in particular T is not MSS-preserving. (cid:3) Remark 5.11.
From Theorem 3 . .
10, we have that theclass of finite MSS-preserving semigroups is contained within the class offinite SSS-preserving semigroups. However, as already observed, there isno such containment between the class of finite WSS-preserving semigroupswith either finite SSS-preserving or finite MSS-preserving semigroups.The following example is a consequence of the proof of Theorem 5 .
10. Itshows that the direct product of two MSS semigroups need not be MSS,even when one of the factors is finite.
Example 5.12.
The direct product of A = h a, b, c | ab c = b i and the twoelement zero semigroup N = { x, } (with the zero element 0) is not mono-genic subsemigroup separable. This follows from the proof of Theorem 5 . N is not a union of groups.Although it is not true that the direct product of an MSS semigroup witha residually finite semigroup is necessarily MSS, if we strengthen the as-sumption of monogenic subsemigroup separability to weak subsemigroupseparability, we obtain a positive result, which we present below. Proposi-tion 5 .
13 can be seen as a successor of Theorem 5 .
10, be it can also be vieweda variation of Theorem 4 .
6. The similarities between Proposition 5 .
13 andTheorem 4 . . Proposition 5.13.
The direct product of a weakly subsemigroup separa-ble semigroup S with a residually finite periodic semigroup T is monogenicsubsemigroup separable.Proof. Let U = h ( s, t ) i ≤ S × T and let ( x, y ) ∈ ( S × T ) \ U . We split intocases. Case 1.
Suppose h s i ≤ S is finite. As T is periodic, we also have that h t i is finite. Therefore U is also finite. As S is MSS it is also residually finite([10, Proposition 2.1]) and therefore S × T is residually finite. Hence we canseparate ( x, y ) from U . Case 2.
Suppose that x / ∈ π S ( U ). Then x / ∈ h s i ≤ S . So we can separate( x, y ) from U by factoring through S and invoking the weak subsemigroupseparability of S . Case 3.
Now suppose that h s i ∼ = N and that x ∈ π S ( U ). Then x = s i forsome i ∈ N . As S is WSS, we can separate s i from h s i +1 , s i +2 , . . . , s i − i = { s k | k ≥ i + 1 } . Hence we can separate ( x, y ) from { ( s k , t k ) | k ≥ i + 1 } by factoring through S and using the weak subsemigroup separability of S . That is, there exists a finite semigroup P and a homomorphism φ : S × T → P such that φ ( x, y ) = φ ( s n , t n ) for all n ≥ i + 1. As S is WSS, itis residually finite by [10, Proposition 2.1]. Hence S × T is residually finiteby [6, Theorem 2]. Then we can separate ( x, y ) from the finite set { ( s k , t k ) | k ≤ i } . That is, there exists a finite semigroup P and homomorphism φ : S × T → P such that φ ( x, y ) = φ ( s k , t k ) for 1 ≤ k ≤ i . Then thehomomorphism φ : S × T → P × P given by φ ( a, b ) = ( φ ( a, b ) , φ ( a, b ))separates ( x, y ) from U . (cid:3) Remark 5.14.
If every residually finite periodic semigroup is WSS, thenProposition 5 .
13 becomes a specific instance of Theorem 4 .
6. However, it isnot known whether every residually finite periodic semigroup is WSS. Thisis left as an open question at the end of this section.We conclude this section, and the paper, with some open problems.
Open Problem 5.15.
Is there a characterisation of when the direct productof two monogenic subsemigroup separable semigroups is itself monogenicsubsemigroup separable?
Open Problem 5.16.
If the direct product of two semigroups is monogenicsubsemigroup separable, is at least one of the factors monogenic subsemi-group separable?
Open Problem 5.17.
Is every residually finite periodic semigroup weaklysubsemigroup separable?
References [1] S. I. Adian. Defining relations and algorithmic problems for groups and semigroups.
Proc. Steklov Inst. Math. , 85:1–152, 1966.[2] R. B. J. T. Allenby and R. J. Gregorac. On locally extended residually finite groups.
Conference on Group Theory . Springer Berlin Heidelberg. 9–17, 1973[3] R. V. Book and F. Otto,
String-rewriting systems , Texts and Monographs in Com-puter Science, Springer-Verlag, New York, 1993.[4] `E. A. Golubov. Finite separability in semigroups.
Sibirsk. Mat. ˇZ. , 11:1247–1263,1970.[5] `E. A. Golubov. The direct product of finitely divisible semigroups. (Russian)
Ural.Gos. Univ. Mat. Zap. , 8:28–34, 1972[6] R. Gray and N. Ruˇskuc. On residual finiteness of direct products of algebraic systems.
Monatsh. Math. , 158:63–69, 2009.[7] M. Hall. Coset representation in free groups.
Trans. Amer. Math. Soc. , 67:421–432,1949.[8] J. M. Howie.
Fundamentals of Semigroup Theory . LMS monographs. Clarendon Press,1995.[9] A.I. Mal’cev. On homomorphisms onto finite groups.
Ivanov. Gos. Ped. Inst. Uˇcen.Zap. , 18:49–60, 1959.[10] C. Miller, G. O’Reilly, M. Quick and N. Ruˇskuc. On separability finiteness conditionsin semigroups. arXiv2006.08499v1, 2020.[11] J. Rhodes and B. Steinberg.
The q-theory of Finite Semigroups . Springer Sciences &Business Media, 2009.
N SEPARABILITY PROPERTIES IN DIRECT PRODUCTS OF SEMIGROUPS 21 [12] P. Stebe. Residual finiteness of a class of knot groups.
Comm. Pure Appl. Math. ,21:563–583, 1968[13] B. de Witt. Residual finiteness and related properties in monounary algebras andtheir direct products. arXiv:2008.07943v1 , 2020.
School of Mathematics and Statistics, University of St Andrews, St Andrews,Scotland, UK
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