Pacific Journal ofMathematics

SUBDIRECTLY IRREDUCIBLE IDEMPOTENT SEMIGROUPS

JAMES ARTHUR GERHARD

Vol. 39, No. 3 July 1971

PACIFIC JOURNAL OF MATHEMATICSVol. 39, No. 3, 1971

SUBDIRECTLY IRREDUCIBLE IDEMPOTENT SEMIGROUPS

J. A. GERHARD

Subdirectly irreducible idempotent semigroups have beendiscussed by B.M. Schein. Using his results, a subdirectlyirreducible idempotent semigroup is shown to be either asemigroup of mappings of a set into itself, with certainstated conditions, or the dual of such a semigroup, or oneof these with an adjoined zero. This characterization isused to show that, with a few exceptions, the join reducibleelements of the lattice of equational classes of idempotentsemigroups contain only subdirectly irreducible membersbelonging to proper subclasses. This result gives structuretheorems, special cases of which appear in the literature. Anexample is also given of an infinite subdirectly irreducibleidempotent semigroup in the equational class of idempotentsemigroups defined by xyx = xy.

The starting point for the study in this paper is the paper ofB. M. Schein [5], in which he gives necessary conditions for anidempotent semigroup to be subdirectly irreducible. This result isexploited in § 1 to give a characterization of subdirectly irreducibleidempotent semigroups (Theorem 1.5).

In §11, the description of the lattice of equational classes ofidempotent semigroups given in [1], and the characterization of sub-directly irreducible idempotent semigroups given in §1 are used toshow that a subdirectly irreducible idempotent semigroup whichsatisfies certain equations also satisfies other, more restrictive equa-tions. These results are used to prove Theorem 2.4 and its corollary.These give conditions under which a join reducible equational classcontains no subdirectly irreducibles not contained in a proper subclass.The other join reducible classes are subclasses of the class defined bythe equation (xyzx = xzyx), and a complete list of the subdirectlyirreducibles of this class is given. (There are only six of these; thecardinality of the largest is three).

Finally we give an example of an infinite subdirectly irreducibleidempotent semigroup which satisfies the equation (xyx = xy). Thisanswers a question of Schein [6] and shows that all equationalclasses of idempotent semigroups except the subclasses of the classdefined by (xyzx = xzyx) contain infinite subdirectly irreducible idem-potent semigroups.

1* The characterization* Let S be a semigroup. If a, be S,let θ(a, b) denote the smallest congruence which identifies a and 6.

669

670 J. A. GERHARD

Let Δ = {(s, s)\seS}. The semigroup S is subdirectly irreducible ifand only if there exist a, be S, a Φ b, such that θ(a, b) gΞ#(e> c£) forall c, de S, c Φ d. The congruence 0(α, b) is just the smallest con-gruence of S which is distinct from A.

The dual (S*9 *) of a semigroup (S, •) is defined by S* = S anda*b = δ α for all a,beS. If we denote a semigroup simply by S,we denote its dual by S*.

A zero of a semigroup is an element 0 e S which satisfies #0 = Ox =0 for all # e S . A semigroup has at most one zero.

B. M. Schein has shown (Theorem 3.6 and Theorem 4.7 of [5]),that in order to characterize subdirectly irreducible idempotent semi-groups, it is sufficient to consider those without zero. This result isgiven in the following theorem.

THEOREM 1.1 (Schein [5]). An idempotent semigroup S whichcontains a zero 0 is subdirectly irreducible if and only if S—{0} is asubsemigroup which is subdirectly irreducible and (if\S\>2) containsno zero.

For the remainder of this section we will deal only with sub-directly irreducible idempotent semigroups without zero.

THEOREM 1.2 (Schein [5], Theorem 4.7). If S is a subdirectlyirreducible idempotent semigroup (without zero), then S satisfies oneof the following conditions.

(1.3) Let K = {k\ks = k for all seS} .

Then K is a two-sided ideal of S, and for any x, y e S, xk = yk forall ke K implies x = y.

(1.3)* Let K={k\sk = k for all seS} .

Then K is a two-sided ideal of S, and for any a?, y e S, kx = ky for allkeK implies x = y.

It is clear that S satisfies (1.3) if and only if S* satisfies (1.3)*.Because of this we will say that condition (1.3)* is the dual of (1.3).In a similar way the following lemma will give rise to a dual lemma(which we will label with*) and which will be obtained just as (1.3)*was obtained from (1.3), that is by replacing the operation everywhereit occurs by the operation*, or more simply by assuming the originalcondition holds for S* instead of S.

LEMMA 1.4. Let R be a subdirectly irreducible idempotent semi-

SUBDIRECTLY IRREDUCIBLE IDEMPOTENT SEMIGROUPS 671

group (without zero) which satisfies (1.3). Let θ(a, b) be the smallestcongruence of S distinct from Δ. Then a e K.

Proof. Let k e K. (K = 0 is impossible by (1.3) since S has nozero and hence | S | > 1.) Since k is not a zero of S and ks = k forall se S, there exists se S such that sk Φ k. It follows that(a, 6) G 0(sfc, fc). Since iΓ is a two-sided ideal, θ(sk, k) S if2, and there-fore ae K.

THEOREM 1.5. An idempotent semigroup S (without zero) is sub-directly irreducible if and only if S or S* is isomorphic to a semi-group T which satisfies the following two conditions:

( i ) C(X) £ Γ g Xx, where Xx is the semigroup of all mappingsof X into itself, C(X) is the set of all constant mappings of X, andT is a subsemigroup of Xx.

(ii) There exists k, k'eC(X) such that θ{k,kf) S 0(c, d) for allc, deC(X), cΦ d.

Proof. Let S satisfy (1.3). Define <p: S -> Kκ by (φ(s))(k) = sk,for all se S, ke K. It is easy to check that φ is a homomorphism,and that φ is one-one. The monomorphism establishes (i) and (ii) forφ(S), since φ{K) — C{K). If S satisfies (1.3)*, the above argumentshows that S* is isomorphic to a semigroup T satisfying (i) and (ii).

To establish the converse, it is enough to show that if T satisfies(i) and (ii), then T is subdirectly irreducible. By (ii) it is enough toshow that if s, te T, s Φ t, then there exist c, de C(X), c Φ d, suchthat θ(c,d)ξiθ(s,t). Since s Φ t, there exists keC(X) such thatsk Φ tk. Since sk, tk e C(X) and θ(sk, tk) £ θ(s, t), the proof is complete.

II* Subdirectly irreducible idempotent semigroups and equa-tional classes*

1. Join reducible classes.The lattice of equational classes of idempotent semigroups has

been completely described in [1]. We refer the reader to that paperfor any notation not explained here, and especially to Fig. 2 of [1],which gives a summary of some of the results which we use. Ourfirst task is to show that a subdirectly irreducible semigroup whichis assumed to satisfy certain equations can then be shown to satisfymore restrictive equations.

LEMMA 2.1. Let S be a subdirectly irreducible idempotent semi-group which satisfies (1.3). If S satisfies (p = q), where for some% ^ 3, I E{p) I = n, p ~ nq, and if n = 3, pR3q, then S satisfies (p(0) =9(0)).

672 J. A. GERHARD

Proof. Without loss of generality, p = p(0)p(0)F(p) and q =g(0)g(0)F(g), (by Theorem 1.8 of [1]). It follows immediately fromProposition 2.13 of [1] if n ^ 4, and from the definition of i?3, if n =3, that p(0) = <z(0). Let ^ be any substitution of the variables oc-curing in (p = q) by elements of S. (As usual ψ may be thoughtof as a mapping of the free semigroup on suitable generators intoS.) Since p = q holds in S we have

In particular if ψ(p(0)) = τKg(0)) = keK, then ψ(p(0))k = f(q(O))Jc.Now p(0) ί ^(^(0)) = 7( (0)), and consequently the above argumentshows that ψ(p(Q))k = ψ(q(0))k for all ke K. It follows from Lemma1.6 that τKp(O)) = iHg(0)) and therefore that p(0) = g(0) holds in 5.

LEMMA 2.2. Lβ£ S be a subdίrectly irreducible idempotent semi-group which satisfies (1.3). If S satisfies an equation (p = q), wherepRnq, pSnq for some n ^ 3, then S also satisfies an equation (p0 = q0)where if n^4, p0Tn^q01 p0Rϊ~iq0, Po£»*-i?o and if n = 3, /(p0) = /(g0)(and therefore S(p0) = S(g0)) α^ώ H*(q0) Φ H*(qQ).

Proof. It is enough to prove that the Lemma holds if in additionpTZq. In this case we can assume without loss of generality thatI E(p) I = n, since (p = q) defines a member of the skeleton of thelattice. We will show that (p(0) = q(0)) can be taken for the equation

(Po = ?o).

By Lemma 2.1 we know that S satisfies (p(0) = (0)). The factthat p(0) and q(0) satisfy the conditions of the present lemma isgiven implicitly in [1]. The rest of the proof consists of showingwhich results of [1] can be used to show that p(0) and q(0) satisfythese conditions.

We consider the case n ^ 4 and n = 3 separately. If n Ξ> 4, itfollows from Proposition 2.13 of [1] that ϊ(p(0)) - ϊ(g(0)) and there-fore in particular that p(0)Tn^q(0). Since pRnq, it follows immedi-ately from (2.20) of [1] that ^(0)22*^(0). If we had p(0)S *_,<?(()),then by (2.17) of [1], it would follow that p(0) - g(0), and thereforethat pSnq. This contradiction shows that ^(0)^1^(0).

If n — 3 the definition of pR3q gives immediately that I(p(0)) —I(g(0)). If H*(p(0)) - H*(q(0)), it follows from Proposition 2.4 of [1]that p(0) — 3g(0). Since \E(p(0))\ = 2, this implies that p(0) ~ g(0),and therefore in particular that pS3q. This contradiction establishesthat jff*(p(O)) Φ H*(q(0)), completing the proof of the lemma.

LEMMA 2.3. Let S be an idempotent semigroup which satisfies

SUBDIRECTLY IRREDUCIBLE IDEMPOTENT SEMIGROUPS 673

(1.3). If S satisfies an equation (p = q) where pRnq for some n^S,then S also satisfies an equation (pQ = q0) where if n ^ 4, PoS^^qoand PoX*+q» and if n = 3, E(pQ) Φ E(qQ)f H(pQ) = H(qo)y H*(p,) ΦH*(q0).

Proof. We deal with the case n = 3 separately since Lemma 2.1cannot be applied in this case. It follows from results of [1] (seePig. 2) that if S satisfies an equation (p = q), where pRzp, it satisfies(xyzx — xzyzx), since this equation satisfies pK3q and pS*q. But thenxyz = xz for all x,yeSy zeKf and therefore, xy = x for all x>yeS.The equation (xy = x) can be taken for the equation (p0 = q0) in casen = 3, since it satisfies the given conditions.

In case n ^ 4, we proceed in a manner similar to that used forthe proof of Lemma 2.2. We may assume in addition that pStq andthat I E(p) I = n, and apply Lemma 2.1 to show that (p(0) — q{ϋ))holds in S. It remains only to show that p(0) and q(0) satisfy thestated conditions.

As in the proof of Lemma 2.2 we can establish that ΐ(p(0)) ~T(q(0)) and therefore in particular that p{0)Sn^q(0). That p(0)^*^(0)is an immediate consequence of 2.20 of [1].

THEOREM 2.4. Let ϊt = % V 2t2 be the join of its proper subclassesSίi and St2, in the lattice of equational classes of idempotent semigroups.If Se$t is subdirectly irreducible without zero, then S e 2^ (J 2C2

Proof. The classes St which satisfy the hypothesis of this theoremcan be obtained from Fig. 2 of [1]. We will prove the theorem foreach in turn. It is of course enough to show that each subdirectlyirreducible S e 21 is contained in a proper subclass of §ί. We list theseveral cases and indicate the results used in each. We give a com-plete proof for the first example but leave the details of the otherexamples to the reader.

Let §1 be defined by an equation (p = q) for which pRnq, p8nq,pR*q and assume n ^ 4. If S satisfies (1.3) we can use Lemma 2.2to show that S satisfies an equation (p0 == #0) where PoΓn-iϊo* VoRt-ιq^PoSί-do. It follows that (p0 = q0) defines a proper subclass of St.If S satisfies (1.3)*, S* satisfies (1.3) and we can use Lemma 2.3 forS* (or Lemma (2.3)* for S) to show that S* satisfies an equation(Po = #o) where poSί»-i?o and po#»-i?o But then S satisfies an equation(p* = 00*) where PoSZ^q* and p*!!*-&*, and (p0* = #o*) therefore definesa proper subclass of 2C. The proof for n = 3 is similar.

If §1 is defined by an equation (p — q) where pRnq, pRϊq, p8nq, p8*q,then if S satisfies (1.3) use Lemma 2.2, and if S satisfies (1.3)* useLemma (2.2)*.

674 J . A. GERHARD

If 3Ϊ is defined by an equation (p = q) where pTnq, pSnq,p8£q, then an equation (pf = qr) where p'Kn+ιq

f also holds in 31. IfS satisfies (1.3) use Lemma 2.3 (in case n + 1), and if S satisfies(1.3)* use Lemma (2.2)*.

If 21 is defined by an equation (p — q) where pKnq, pK%q then ifS satisfies (1.3) use Lemma 2.3, and if S satisfies (1.3)* use Lemma(2.3)*.

If 3ί is defined by an equation (p = q) where E(p) = E(q), H(p) =H(q), I(p) Φ J(g), H*(p) Φ H*(q), then an equation (pf = q') wherep'Kfq* and ptRzq

t also holds in St. If S satisfies (1.3) use Lemma 2.3 toshow that S satisfies an equation (p0 = q0) where E(p0) Φ E(q0), H(p0) =H(qQ), H*(pQ) Φ H*(qQ). If S satisfies (1.3)* use Lemma (2.3)* to showthat S satisfies an equation (p0 = q0) where E(p0) Φ E(q0), H(p0) Φ H(q0)and H*(po) = H*^). Since S satisfies (p0 = q0) and (p = q) it alsosatisfies the equation (x ~ y) which defines a proper subclass of 31.

The remaining cases are just the duals of the above.

COROLLARY 2.5. Let 31 = % V 3t2, where % and % are definedby equations (/L = gj and (/2 = g2) where E(f^) — E(g^ and E(f2) =E(g2). If Se3ί is subdirectly irreducible then Se3t1U2ί2. (E(f) isthe set of variables of /.)

Proof. The theorem establishes the result in case S does notcontain a zero. By Theorem 1.1, the subdirectly irreducible semi-groups with zero are obtained by adding a zero to a semigroup with-out zero. It is easy to see that if S — S U {0} is a semigroup suchthat S is a subsemigroup, the equations (/ = g) true in S are exactlythese true in S with E(f) = E(g), and this establishes the corollary.

REMARK 1. Corollary 2.5 is equivalent to the statement that for3t, 3ίL and 3ί2 as given, if S e % then S is a subdirect product ofSx e Sti and S2 e 3I2. This result can of course be proved withoutreference to subdirectly irreducibles. In fact the solutions of thevarious word problems, which show that f — g holds in 3t if andonly if a condition holds for /(/) and /(#), and a second conditionholds for F(f) and F(g), enable us to define congruences θy and θ2,on any free algebra T in 3ί, such that θ1 A θ2 = A. The splitting ofnon-free algebras is accomplished via the appropriate isomorphismtheorem by showing that {θ1 V θ) Λ (θ2 V θ) — θ for any congruenceθ on T. This last result is easy to check since by definition if fθxuand fθ2v, then f = uv in T. Taking (/, g) e {θι V θ) A (θ2 V 0), andusing this result enables us to show immediately that (/, g) eθ. Weleave the details to the interested reader.

SUBDIRECTLY IRREDUCIBLE IDEMPOTENT SEMIGROUPS 675

REMARK 2. Special cases of the above results are known ([2],[3], [4], [7]) In particular the paper of M. Petrich [3] gives theresults of our next section. We include the next section, however,for completeness and because the arguments become especially simplein view of Theorem 2.4 and its Corollary.

2. Subdirectly irreducible semigroups in subclasses of the classdefined by the equation (xyzx — xzyx).

The hypotheses of Corollary 2.5 hold for all join reducible elementsof the lattice of equational classes of idempotent semigroups exceptthose which are proper subclasses of the class defined by (xyzx = xzyx).The conclusion of Corollary 2.5 holds for the class defined by (xyx —x) but fails for the class defined by (xyz = xzy) and the class definedby its dual. To show this we list all subclasses of the class definedby (xyzx = xzyx), together with their subdirectly irreducible elements.

In the class defined by (x = y) the only (subdirectly irreducible)member is the semigroup So consisting of one element. In the classdefined by (xy = x), the subdirectly irreducible members are So, and S19

the two element semigroup which satisfies (xy — x). In the class definedby (xy — yx), there can be no subdirectly irreducible semigroup satis-fying (1.3) or (1.3)* since (by Lemma 1.4), \K\ ^ 2, and for k, Jc'eK,k Φ k\ kkf — k Φ kr = k!k. The subdirectly irreducible elements inthis class are therefore just So and So U {0}. By our general results,the subdirectly irreducible elements of the class defined by (xyz =xzy) are So, S0U {0}, St and the semigroups formed from these byadding a zero if the semigroup has no zero. This gives the newsemigroup St U {0}. In the class defined by (xyx — x) the only sub-directly irreducible elements are Slf S* and So since E(xyx) Φ E(X).Finally the subdirectly irreducible elements of the class defined by(xyzx = xzyx) are just SQ, SQ U {0}, Sl9 St U {0}, Sf, S? U {0}, The re-maining subclasses of the class defined by (xyzx = xzyx) are the dualsof classes already listed.

3. An infinite subdirectly irreducible idempotent semigroup satis-fying (xyx = xy).

In the previous section we listed several equational classes ofidempotent semigroups with only finite subdirectly irreducible ele-ments. In this section we show that every other equational classhas infinite subdirectly irreducible members by exhibiting an infinitesubdirectly irreducible idempotent semigroup satisfying (xyx = xy).This answers a question of B. M. Schein [6].

Let S consist of the following mappings of the set of naturalnumbers into itself. The mappings a{, ί ^ 0 are the constant mappingsdefined by a{(j) — i for all j . The mappings bif c<, i ^ 2, are defined

676 J. A. GERHARD

by bi(ΐ) = Ci(ΐ) = i, bi(j) = 0, ί ^ i, c<(j) = l,iφ j . Since the α<f δ< andC{ are idempotent, we can show that S is an idempotent semigroupby proving that it is closed with respect to composition of mappings.If one of the factors is a constant the result is a constant. If bothare non-constants, they occur among the following: b&i = b{, cfii = cζ

and if i Φ j , bfij = bicύ = α0, c&j = c<6y = αx.In order to show that S is subdirectly irreducible it is enough

(by Theorem 1.5) to show that θ(a0, aλ) S θ(aif a3) for all i Φ j . If i Φj , θ(aif a,j) contains (α< = bi<Li9 a0 = 6<ay) and (a{ — c^, ay — c<αy) and

therefore (α0, αx).We now show that S satisfies (xyx = xy). If x, y or xy is a con-

stant, or if x = y, the result is trivial. The only other case is {x, y} —{bi, cj for some i, and in that case it is easy to check that xyx — xy.

REFERENCES

1. J. A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra,15 (1970), 195-224.2. N. Kimura, The structure of idempotent semigroups (I), Pacific J. Math., 8 (1958),257-275.3. M. Petrich, Homomorphisms of a semigroup onto normal bands, Acta Sci. Math.Szeged, 27 (1966), 185-196.4. B. M. Schein, A contribution to the theory of restrictive semigroups, Izv. vys. uc. zav.Mat. 33 (1963), 152-154 (in Russian).5. , Homomorphisms and subdirect decomposition of semigroups, Pacific. J.Math., 17 (1966), 529-547.6. , Subdirectly irreducible infinite bands: An example, Proc. Japan Acad., 42(1969), 1118-1119.7. M. Yamada, A note on subdirect decompositions of idempotent semigroups, Proc.Japan Acad. 36 (1960), 411-414.

Received February 23, 1971 and in revised form June 11, 1971. The results of thispaper were announced in abstract 71T-A67 Notices America Math. Soc. This researchwas supported by the National Research Council of Canada.

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Pacific Journal of MathematicsVol. 39, No. 3 July, 1971

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d’ensembles convexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641Leslie R. Fletcher, A note on Cθθ -groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655Humphrey Sek-Ching Fong and Louis Sucheston, On the ratio ergodic theorem for

semi-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659James Arthur Gerhard, Subdirectly irreducible idempotent semigroups . . . . . . . . . . . . . 669Thomas Eric Hall, Orthodox semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677Marcel Herzog, Cθθ -groups involving no Suzuki groups . . . . . . . . . . . . . . . . . . . . . . . . . 687John Walter Hinrichsen, Concerning web-like continua . . . . . . . . . . . . . . . . . . . . . . . . . . 691Frank Norris Huggins, A generalization of a theorem of F. Riesz . . . . . . . . . . . . . . . . . . . 695Carlos Johnson, Jr., On certain poset and semilattice homomorphisms . . . . . . . . . . . . . 703Alan Leslie Lambert, Strictly cyclic operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 717Howard Wilson Lambert, Planar surfaces in knot manifolds . . . . . . . . . . . . . . . . . . . . . . 727Robert Allen McCoy, Groups of homeomorphisms of normed linear spaces . . . . . . . . 735T. S. Nanjundiah, Refinements of Wallis’s estimate and their generalizations . . . . . . . 745Roger David Nussbaum, A geometric approach to the fixed point index . . . . . . . . . . . . 751John Emanuel de Pillis, Convexity properties of a generalized numerical range . . . . . 767Donald C. Ramsey, Generating monomials for finite semigroups . . . . . . . . . . . . . . . . . . 783William T. Reid, A disconjugacy criterion for higher order linear vector differential

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795Roger Allen Wiegand, Modules over universal regular rings . . . . . . . . . . . . . . . . . . . . . . 807Kung-Wei Yang, Compact functors in categories of non-archimedean Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821R. Grant Woods, Correction to: “Co-absolutes of remainders of Stone-Cech

compactifications” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827Ronald Owen Fulp, Correction to: “Tensor and torsion products of

semigroups” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827Bruce Alan Barnes, Correction to: “Banach algebras which are ideals in a banach

algebra” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

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