The Isomorphism Problem for Orthodox Semigroups

Pacific Journal of Mathematics

THE ISOMORPHISM PROBLEM FOR ORTHODOX SEMIGROUPS

THOMAS ERIC HALL

Vol. 136, No. 1 November 1989 PACIFIC JOURNAL OF MATHEMATICS Vol. 136, No. 1, 1989

THE ISOMORPHISM PROBLEM FOR ORTHODOX SEMIGROUPS

T. E. HALL

The author's structure theorem for orthodox semigroups produced

an orthodox semigroup %?(Et T, ψ) from a band £, an inverse semigroup T and a morphism ψ between two inverse semigroups, namely T and WΈIyi an inverse semigroup constructed from E. Here, we solve the isomorphism problem: when are two such orthodox semigroups isomorphic? This leads to a way of producing all orthodox semigroups, up to isomorphism, with prescribed band E and maximum inverse semigroup morphic image T.

1. Preliminaries. A semigroup S is called regular (in the sense of von Neumann for rings) if for each a e S there exists x e S such that axa = a; and S is called an inverse semigroup if for each a e S there is' a unique x £ S such that axa = a and xax = x. A band is a semigroup in which each element is idempotent, and an orthodox semigroup is a regular semigroup in which the idempotents form a subsemigroup (that is, a band). We follow the notation and conventions of Howie [4].

Result 1 [3, Theorem 5]. The maximum congruence contained in Green's relation %? on any regular semigroup S, μ = μ(S) say, is given by μ = {(a,b) e 2P\ for some [for each pair of] 2?'-related inverses d of a and b1 ofb, a'ea — b'eb for each idempotent e < aa1}.

A regular semigroup S is called fundamental if μ is the identity relation on S. For each band E, the semigroup WE is fundamental, orthodox, has its band isomorphic to E, and contains, for each orthodox semigroup S with band E, a copy of S/μ as a subsemigroup: see the author [1] (or [3] with E = (E) and WE = T{E)) or Howie [4, §VL2]. Now take any inverse semigroup Γ, and, if such exist, any idempotent-separating morphism ψ: T —• WE/γ whose range contains the semilattice of all idempotents of WEjy, where γ denotes the least inverse semigroup congruence on WE. A semigroup β?(E, T, ψ) (see

123 124 T. E. HALL

S(E, T, ψ) in the author [2], or see Howie [4, §VL4]) is defined by

jr(E, T, ψ) = {{w, t)eWEx T: wγ* = tψ}\ that is, J?{E, T, ψ) occurs in the pullback diagram

JT{E,T,ψ) —^ T

WE > WE/γ

Here, p\ and p2 are projections. The semigroup β?{E, T, ψ) is orthodox, has band isomorphic to E, and has its maximum inverse semigroup morphic image isomorphic to T; conversely every such semigroup is obtained in this way (the author [2], or Howie [4, §VL4]).

2. The isomorphism problem.

LEMMA 1. Take any two morphisms φ, ψ from a regular semigroup T to a regular semigroup S such that the range of each of φ and ψ contains the set E(S) of all the idempotents ofS. Ifφ\E(T) = ψ\E{T) then (tφ, tψ) e μ, for all t e T; in particular, if also S is fundamental then φ = ψ.

Proof. Take any t G T and any inverse t' of t in T. Of course, in S, t'φ and t'ψ are inverses of tφ and tψ respectively and (t'φ)(tφ) = (t't)φ = {t't)ψ = (tfψ)(tψ). Likewise (tφ)(t'φ) = (tψ)(t'ψ), so (tφ)^(tψ) and (t'φ)^(t'ψ). Take any idempotent e of S such that e < (tt')φ and any x e T such that xφ = e: then (tt'xtt')φ = [(tt')φ]e[(tt')φ] = e, so e e range(φ\tt'Ttt'). Now tt'Ttt' is a regular semigroup, so by Lallement's Lemma [4, Lemma II.4.7] there is an idempotent / e tt'Ttt' such that fφ = e. Since t1 ft is idempotent, we have {t'φ)e{tφ) = {t1 φ){fφ){tφ) = (t'ft)φ = (t'ft)ψ = (t'ψ)(fψ)(tψ) = (t'ψ)e(tψ). Thus {tφ, tψ) G μ, as required, completing the proof.

Take any isomorphism α: E —• E' from a band E to a band E1. Con- sider WE and WEι and, as usual, identify E and E' with the bands of THE ISOMORPHISM PROBLEM FOR ORTHODOX SEMIGROUPS 125

1 WE and WE> respectively. Since WE> is constructed from E precisely as WE is constructed from E, there is an isomoφhism from WE to WE> extending α, say a* (in fact, there is a unique such isomoφhism, by Lemma 1). Denote by γ and / the least inverse semigroup congruences on WE and WE, respectively: then the map α**: WE/γ -> WE>/γ', given by wγa** = wa*γ\ for all w e WE, is an isomoφhism such that γ^a** = a*γ'\ and is the unique such isomoφhism. Summarizing, we have that the following diagram commutes, and a*, a** are the unique moφhisms making the diagram commute.

1 WE/y -2CL, Wvly

THEOREM 2. Take any bands E, E\ inverse semigroups T, V and 1 idempotent-separatingmorphisms ψ: T —• WΈ/γ andψ : V —»• WE'/y' whose ranges contain the idempotents of WE I y and WEΊY1 respectively.

Then MT{E, Γ, ψ) is isomorphic to β?(E'y V, ψ') if and only if there exist isomorphisms a: E -> E and β:T -* V such that the following diagram commutes

WEIJ -^ WE,/γ' that is, such that ψ' = β~xψa**.

Proof, (a) if statement. Suppose such a, β exist. Informally we could say that E', V, ψ' are a renaming of E, T, ψ respectively, obtained by renaming each e € E by ea and each / e T by tβ, and so SfT{E', V, ψ') is isomorphic to &(E, T, ψ). More formally, we consider the isomorphism (a*,β): WE x T —*• WΈ x T' given by

(w, t)(a*. β) = (u/α , tβ) for all (w, t) e WE x T, and we show that JT(E, T, ψ){a*. β) = βT(E', V, ψ'). 126 T. E. HALL

Take any (w, t) e &(E, T, ψ): then wγ* = tψ, and so

tβψ' = tββ~ιψa** = tψa** = wγ^a** = wa*γ*, so (w, t)(a*,β) = (wa*, tβ)eJ^(Ef, V, ψ') and hence *(E, T, ψ)(a*,β) c &{E't V, ψ'). From symmetry, we deduce that

', T', ψ'){a\ β)~ι = X{E\ V, ^(α*"1 * ^-1) £ ^(^, Γ, ^),

! whence JF(£, Γ, ψ)(a*, β) = βf{E'y V, ψ), as required. (b) only if statement. Informally, we could say that, for any orthodox semigroup S with band E and least inverse semigroup congruence y, there is a unique morphism ψ making the following diagram commute:

s -^u wE \ V Ψ — -^ WE/y

Hence E, S/p', ψ are all determined to within isomorphisms (or re- namings) by S. Formally, we proceed as follows. Take any isomorphism θ: S -» S', where S = βT{E,T,ψ) and S' = %f{E', V, ψ'). Put Θ\E = a, an isomorphism of E upon E', by Lallement's Lemma [4, Lemma II.4.7]. Let y and %" denote the least inverse semigroup congruences on S and S' respectively. Clearly there is a unique isomorphism β: S/y —• S'/y making the following diagram commute:

S —θ—+ S'

s/y -J-* s /y

Now T = S/y and V = S'/y' ([2, Theorem 1] or [4, Theorem VI.4.6]), so we assume without loss of generality that T = S/y and V = S'/y; it remains to show that ψ' = β~ιψa**. THE ISOMORPHISM PROBLEM FOR ORTHODOX SEMIGROUPS 127

We shall see that Diagram 1 commutes (p\, pi, ρ\, p'2 are projections).

L=(/U) U=(/>'Λ#

WE/y DIAGRAM 1 We have seen already that each of the four outer faces is a commuting diagram: we consider the central face. Now θp[ and p\a* are morphisms which agree on E (with a = θ\E), and which map E (isomor- phically) onto 2s', the band of W&. Hence, by Lemma 1, θp[ = p\a*\ that is, the central face commutes. Consideration of the external face leads us to the following diagram.

WE ••=4

The commuting of the five internal faces of Diagram 1 gives us that p\j* = P2βψ'a**~ι. But the mapping sψ v-+ (ps,λs)γ (for all s G S), namely ψ, is the unique morphism from T to WE/γ making this diagram commute, and hence ψ = βψfa**~ι (that is, the external face commutes) and so ψ' = β~ι ψa** as required. 3. Orthodox semigroups, up to isomorphism. Consider the following problem: given a band E and an inverse semigroup Γ, find, up to isomorphism, the orthodox semigroups with band E and with maximum inverse semigroup morphic image isomorphic to T. The author's structure theorem ([2, Theorem 1] or [4, Theorem VI.4.6]) and Theorem 2 above together immediately yield a solution as follows. 128 T. E. HALL

Denote by Aut(S) the group of automorphisms of any semigroup

S. From Lemma 1, for any φ e AΛXX{WE), we see that φ = (φ\E)*9 so we have that Aut(JE) = Aut(WE) under the map a »-> a* for each α e Aut(E). The map Aut(W^) -> Aut(^/y),α* »-> α** (for each a G Aut(is)), is a morphism; we denote its image by [Aut(is)]**; then [Aut(E)]** = {α**: a G Aut(£)}. Denote by M the set of idempotent-separating morphisms from

T into WEjγ whose ranges each contain the idempotents of WE/y. By [2, Corollary 1] or [4, Theorem VI.4.6], there exists an orthodox semigroup with band E and with maximum inverse semigroup morphic image isomorphic to Γ, if and only if M is nonempty. Assume henceforth that M is nonempty. Define an action on M by the group Aut(Γ) x [Aut(£)]** as follows: for all ψ G M, β G Aut(Γ), a G Aut(£). The orbits of M under Aut(Γ) x [AutCE')]** are the sets ψ(Aut(T) x [Aut(E)]**) = {β~ιψa**: β G Aut(Γ), a G Aut(£)}, for each ψ G M (thus these sets partition Af). By Theorem 2, we have, 1 for all ψ9 ψ G M, ^T(£, Γ, ^) = ^(£, Γ, ψ') if and only if ψ and ^ are in the same orbit. Thus, if {ψi•: i G /} is a transversal of the set of orbits (that is, a selection of precisely one morphism from each orbit) then X^(E9 T, ψi), i G /, is a list of all the orthodox semigroups with band E and maximum inverse semigroup morphic image isomorphic to Γ, and the semigroups are pairwise nonisomorphic.

REFERENCES

[1] T. E. Hall, On orthodox semigroups and uniform and antiuniform bands, J. Al- gebra, 16(1970), 204-217. [2] , Orthodox semigroups, Pacific J. Math., 39 (1971), 677-686. [3] , On regular semigroups, J. Algebra, 24 (1973), 1-24. [4] J. M. Howie, An Introduction to Semigroup Theory, L.M.S. Monographs, No. 7, Academic Press, London, New York, 1976.

Received July 21, 1987.

MONASH UNIVERSITY CLAYTON, VICTORIA 3168 AUSTRALIA PACIFIC JOURNAL OF MATHEMATICS EDITORS V. S. VARADARAJAN R. FINN ROBION KIRBY (Managing Editor) Stanford University University of California University of California Stanford, CA 94305 Berkeley, CA 94720 Los Angeles, CA 90024 HERMANN FLASCHKA C. C. MOORE HERBERT CLEMENS University of Arizona University of California University of Utah Tucson, AZ 85721 Berkeley, CA 94720 Salt Lake City, UT 84112 VAUGHAN F. R. JONES HAROLD STARK THOMAS ENRIGHT University of California University of California, San Diego University of California, San Diego Berkeley, CA 94720 La Jolla, CA 92093 La Jolla, CA 92093 STEVEN KERCKHOFF Stanford University Stanford, CA 94305

ASSOCIATE EDITORS R. ARENS E. F. BECKENBACH B. H. NEUMANN F. WOLF K. YOSHIDA (1906-1982) SUPPORTING INSTITUTIONS UNIVERSITY OF ARIZONA UNIVERSITY OF OREGON UNIVERSITY OF BRITISH COLUMBIA UNIVERSITY OF SOUTHERN CALIFORNIA CALIFORNIA INSTITUTE OF TECHNOLOGY STANFORD UNIVERSITY UNIVERSITY OF CALIFORNIA UNIVERSITY OF HAWAII MONTANA STATE UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF NEVADA, RENO UNIVERSITY OF UTAH NEW MEXICO STATE UNIVERSITY WASHINGTON STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF WASHINGTON aicJunlo ahmtc 99Vl 3,N.1 No. 136, Vol. 1989 Mathematics of Journal Paciﬁc Paciﬁc Journal of Mathematics Vol. 136, No. 1 November, 1989

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