opinion: SNEED, Circuit Judge:
  This is an appeal from the district court’s enforcement of a labor arbitration award. We affirm.
  I.
  FACTS
  This case arises from the construction in 1981-82 of a shopping center in Fresno, California. Two general contractors worked on the project, North American Investments (NAI) and Gateway Structures, Inc. (Gateway). Both of these corporations are controlled, though not wholly owned, by John Langford. NAI has never signed a labor agreement with any union. Gateway signed an agreement with the Carpenters Union (the Union) in July 1981. John Langford also signed that agreement; it is not clear whether he signed it as an individual or as a representative of Gateway. His contractor license number appears on the agreement. That agreement incorporates by reference the 1980 Master Agreement between the Northern California Contractors Council, Inc., and the Carpenters 46 Northern California Counties Conference Board of the United Brotherhood of Carpenters and Joiners (the Master Agreement).
  The pension trust funds operated by the Union financed construction of Von’s Market, one of the stores in the shopping center. The terms of that financing required that Union labor be used in the construction of Von’s Market. NAI, the general contractor on the entire project, did not want to sign a labor agreement. Accordingly, it formed Gateway to build Von’s Market and to enter into an agreement with the Union.
  During the same period of time, NAI also was involved in negotiations with the Union. NAI agreed to enter into a collective bargaining agreement with the Union if the Union’s pension trust funds would make a takeout (or permanent) loan on the shopping center. That financing never materialized; NAI never signed any agreement with the Union. NAI did, however, use some Union labor to construct its portion of the shopping center.
  The facts out of which the grievance arose are in dispute. A few things are clear, however. Langford did not maintain a formal separation between Gateway and NAI. Trust fund contributions for employees of one company were paid by the other. Paychecks for employees of one company were written on accounts of the other company. Most importantly, employees dispatched by the Union to one company performed work for the other company.
  On August 17, 1981, the Union filed a grievance against NAI and Gateway claiming that NAI and Gateway were using nonunion labor to build the shopping center. The parties admit that NAI was using nonunion labor. The Master Agreement contained a union security clause, which prohibits employers bound by the agreement from using nonunion labor. The Union claimed that Gateway and NAI should be treated as one entity and accordingly that the agreement required both companies to use only Union labor. Under the terms of the agreement, the dispute was submitted to arbitration.
  Before the arbitration hearing, two proceedings were filed before the NLRB. First, on August 25, the Union filed an unfair labor practice charge. Second, Gateway filed a unit clarification petition. Gateway asked the arbitrator to defer to the NLRB. The arbitrator refused. After a hearing at which the parties introduced considerable evidence about the relationship between Gateway and NAI, the arbitrator found in favor of the Union. Gateway appealed to the District Court. Chief Judge Peckham granted summary judgment for the Union on a counterclaim for enforcement of the arbitration award. On appeal here, Gateway raises two claims: first, that the arbitrator should not have decided the merits of the dispute; and second, that we should not enforce the arbitrator’s decision.
  II.
  DISCUSSION
  A. Arbitrability
  Gateway argues that it was improper for the arbitrator to decide this case for two reasons. First, it claims that the issue was not submitted to the arbitrator. Second, it claims that arbitrators do not have power to decide unit representation issues. Neither reason has merit.
  1. Submission to Arbitration. We commence with a point that is beyond dispute: an issue is arbitrable only if the parties have agreed to arbitration. E.g., George Day Construction Co. v. United Brotherhood of Carpenters, 722 F.2d 1471, 1474-75 (9th Cir.1984). Section 51 of the Master Agreement, which provides for arbitration, applies to “[a]ny dispute concerning the relationship of the parties [and] any application or interpretation of this Agreement.” Excerpt of Record at 20 [hereinafter cited as E.R.]. Interpreting similar language in a collective bargaining agreement, the Supreme Court has said, “There is nothing to limit the sweep of this language or to except any dispute or class of disputes from arbitration. In that circumstance, we must conclude that the parties meant what they said — that ‘any difference’ ... should be referred to the arbitrator for decision.” International Union of Operating Engineers v. Flair Builders, Inc., 406 U.S. 487, 491, 92 S.Ct. 1710, 1712, 32 L.Ed.2d 248 (1972).
  In this case, the Union relies on section 9 of the Master Agreement, which provides:
  This Agreement is binding upon each individual employer regardless of whether or not he or it changes the name or style or address of his or its business.
  Each individual employer, corporate or other legal entity, or its successor, shall be liable under, subject to, and bound by this Agreement.
  E.R. at 10 (emphasis added). The Union asserts that this language binds NAI to the Master Agreement and, thus, to section 51 of the Master Agreement, the arbitration clause. It is clear that this assertion, when rejected by NAI, constitutes a “dispute” within the meaning of section 51 of the Master Agreement. Accordingly, the issue was arbitrable.
  2. The Power of Arbitrators. A bit of background is necessary to explicate Gateway’s claim that arbitrators do not have power to decide representation issues. The Union’s effort to apply the agreement with Gateway to NAI, a nonsignatory employer, can prevail under either of two doctrines — the “alter ego” doctrine or the “single employer” doctrine. Under the alter ego doctrine, the court considers the interrelation of operations, common management, centralized control of labor relations, and common ownership. If these factors show that the transaction is a sham designed to avoid the obligations of a collective bargaining agreement, the nonsignato-ry employer will be bound. See Carpenters’ Local Union No. 1478 v. Stevens, 743 F.2d 1271, 1276-77 (9th Cir.1984). Under the single-employer doctrine, the same factors are considered. To bind the nonsig-natory under this doctrine, however, the two employers must be shown to constitute a single bargaining unit. See id. at 1276.
  It is this requirement upon which Gateway grounds its argument. It points to Brotherhood of Teamsters v. California Consolidators, Inc., 693 F.2d 81 (9th Cir.1982) (per curiam), in which this court held that a district court has jurisdiction to decide single-employer issues but does not have jurisdiction to decide the bargaining unit issue in such cases, id. at 83; see South Prairie Construction Co. v. Local No. 627, International Union of Operating Engineers, 425 U.S. 800, 96 S.Ct. 1842, 48 L.Ed.2d 382 (1976) (per curiam) (holding that federal courts cannot make initial findings on unit-representation issues, but must defer to the NLRB). It follows, Gateway claims, that California Consolidators precludes arbitrators from imposing the obligations of labor agreements on nonsigna-tory employers like NAI.
  We disagree. First, the arbitrator’s decision could rest on the alter ego doctrine, which does not require a unit-representation determination. Cf. Northwest Administrators, Inc. v. Con Iverson Trucking, Inc., 749 F.2d 1338, 1340 (9th Cir.1984) (holding that a district court has jurisdiction over alter ego claims). Second, it probably would be permissible for the arbitrator to make a unit-representation determination even though the district court would be precluded from making such a determination in the first instance. See Carey v. Westinghouse Electric Corp., 375 U.S. 261, 266-72, 84 S.Ct. 401, 406-09, 11 L.Ed.2d 320 (1964) (holding that arbitrators can decide certain representational issues); Stevens, 743 F.2d at 1278-79 & n. 11 (reconciling Carey and South Prairie).
  Accordingly, we hold that it was proper for the arbitrator to proceed to decision of this dispute.
  B. Enforcement of the Arbitrator’s Decision
  Finally, Gateway argues that we should refuse to enforce the arbitrator’s decision. The law by which we must be guided is clear:
  Where the decision involves contractual interpretation, we must defer as to any decision which draws its essence from the agreement. Therefore, if on its face, the award represents a plausible interpretation of the contract, judicial inquiry ceases and the award must be enforced. This remains so even if the basis for the decision is ambiguous and notwithstanding the erroneousness of any factual findings or legal conclusions, absent a manifest disregard of the law.
  However, we are not bound to defer to an award which actually violates the law or any explicit, well defined and dominant public policy.
  George Day Construction Co. v. United Brotherhood of Carpenters, 722 F.2d 1471, 1477 (9th Cir.1984) (citations omitted); see Carpenters’ Local Union No. 1478 v. Stevens, 743 F.2d 1271 (9th Cir.1984) (refusing to enforce an arbitration award that conflicted with the NLRB’s resolution of the same controversy). Gateway suggests four defects in the arbitration award, none of which are sufficient to preclude enforcement of the award.
  First, Gateway asserts that the arbitrator’s decision should not be enforced because the arbitrator did not apply NLRB case law on the single-employer or alter ego doctrines. Gateway tries to bolster this claim by pointing to section 10 of the Master Agreement, which provides that “interpretation of ... this Agreement is ... intended to apply no broader [sic] than that permitted by law.” E.R. at 11. We are unpersuaded. First, as appellant’s own brief adequately demonstrates, the record contained evidence that could have persuaded the arbitrator that the NLRB requirements were satisfied. See Opening Brief for Petitioner/Appellant Gateway Structures, Inc. at 16-22. More importantly, however, George Day makes it clear that parties who bargain for arbitration do not bargain for an award identical to judicial determination. To assert that the arbitrator misapplied a fact-specific test like the alter ego doctrine does not justify a refusal to enforce the arbitration award.
  Second, Gateway argues that enforcement would interfere with the rights of NAI employees by subjecting them to a collective bargaining agreement without any showing of majority support or a determination of the appropriate representational unit. No such inquiry is required. As the Supreme Court explained in Carey v. Westinghouse Electric Corp., 375 U.S. 261, 84 S.Ct. 401, 11 L.Ed.2d 320 (1964), “the possibility of conflict [with a subsequent NLRB ruling] is no barrier to resort to a tribunal other than the Board.” Id. at 272, 84 S.Ct. at 409. If such an NLRB ruling comes,
  the Board’s ruling would, of course, take precedence; and if the employer’s action had been in accord with that ruling, it would not be liable for damages under § 301.
  ... The superior authority of the Board may be invoked at any time. Meanwhile the therapy of arbitration is brought to bear in a complicated and troubled area.
  Id. Although this circuit has never specifically adopted this reasoning, see Stevens, 743 F.2d at 1280 & n. 12 (noting the issue), other circuits have, see id. at 1280 n. 12. We agree that it would be improper to allow the speculative possibility of subsequent NLRB decisions to prevent enforcement of the arbitrator’s 'award against Gateway.
  Third, Gateway contends that we should not enforce the arbitration award on the ground that the arbitrator did not explain his result with sufficient specificity. At the arbitration hearing, the parties agreed that the arbitrator should issue an award promptly and prepare a full opinion later. In the district court proceedings, both parties agreed to continue the case until the arbitrator provided the full opinion. Finally, the arbitrator provided a statement that he adopted his original four-page award as his full opinion.
  We reject Gateway’s contention. The arbitrator’s opinion fairly outlines the basis of his decision — NAI and Gateway operated their companies as essentially one business. Absent a specific provision in the agreement of the parties, we think it is irrelevant that the opinion does not, for instance, discuss the relevant NLRB case law. See George Day Construction Co. v. United Brotherhood of Carpenters, 722 F.2d 1471, 1477 (9th Cir.1984) (noting that a court must enforce an arbitration award even if “the decision is ambiguous and notwithstanding the erroneousness of any factual findings or legal conclusions” (citations omitted)).
  Finally, Gateway claims that the decision does not comport with the terms of its memorandum agreement with the Union. Although there was conflicting testimony, see, e.g., Supplemental Excerpt of Record at 32, 54, we will assume arguendo that the parties agreed that Gateway alone would be bound by the agreement and that Gateway would work only on Yon’s Market. Unfortunately, this assumed parol agreement is not clearly reflected in the written agreement, which does not define the scope of work to be covered. E.R. at 7. Moreover, the written memorandum agreement, like the Master Agreement, refers to the individual employer and the name and style under which he conducts business. It would be reasonable for the arbitrator to conclude that this language referred to John Langford (who signed the agreement and controlled both NAI and Gateway) rather than to Gateway, and that Langford conducted business under the names NAI and Gateway. Langford’s signature on the agreement supports this construction. He did not clearly indicate that he signed only in a representative capacity. His signature could be construed as that of a second individual employer. See E.R. at 7. In brief, although the basis for the arbitrator’s decision is unclear, we hold it to be a plausible decision. It is not our place to substitute our judgment for that of the arbitrator; the parties bargained for his decision, not ours.
  As Chief Judge Peckham pointed out, “The arbitrator’s conclusion did not directly conflict with any clause of the 1981 agreement.” E.R. at 70. Accordingly, the district court’s conclusion that the arbitration award should be enforced is affirmed.
  AFFIRMED.
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