We write to untangle some complications of appellate jurisdiction presented by the union's motion to dismiss the employer's appeal.

The employer fired one of its employees because he had sexually harassed--indeed, sexually assaulted--a female employee. The union took the matter to arbitration on behalf of the fired employee, and the arbitrator, while not denying that the assault had occurred, thought discharge too harsh a remedy and ordered the employee reinstated with back pay to be determined at a subsequent hearing. The employer brought suit under section 301 of the Taft-Hartley Act to set aside the arbitrator's award as contrary to public policy. The district judge rejected the employer's argument, confirmed the arbitrator's award, and returned the matter to the arbitrator for the hearing on the amount of back pay due. The court also turned down the union's request for attorney's fees.

The employer filed a notice of appeal, and the union cross-appealed from the denial of attorney's fees. Shortly afterward the employer rehired the employee in question and on the same day fired him again. The union asked the district court to hold the employer in contempt, but the judge refused to do this, on the ground that the filing of the appeal had divested him of jurisdiction. The union then filed with us this motion to dismiss the employer's appeal, hoping by this means to clear the way for the district judge to consider the motion for contempt.

The ground of the motion to dismiss the employer's appeal is that the order confirming the arbitrator's award is not final, since the matter of back pay remains to be determined by him. It is final in the district court, but under our cases that may not be good enough. Provided that the matter left for determination is not merely ministerial, a matter for mechanical computation--and the determination of back pay is not such a matter, at least on the facts of this case so far as we are able to determine--an order that does not determine the entire substantive relief to which the plaintiff is entitled is not a final decision even if the loose ends remain to be tied up by a decision-maker other than the district judge. United Steelworkers of America v. Aurora Equipment Co., 830 F.2d 753 (7th Cir.1987); Shearson Loeb Rhoades, Inc. v. Much, 754 F.2d 773, 776-77 (7th Cir.1985).

But final decisions do not exhaust the domain of appealable orders, even in the federal courts, strongly committed though they are to the final-order doctrine. Injunctions are appealable without regard to finality, 28 U.S.C. Sec. 1292(a)(1), and this is true of mandatory injunctions--orders to do--as well as of the more common negative injunctions, which are orders not to do, or to stop doing. Schulner v. Jack Eckerd Corp., 706 F.2d 1113 (11th Cir.1983) (per curiam). An order to reinstate an employee is a mandatory injunction, and it was such an order that the district judge entered when he confirmed (that is, enforced) the arbitration award; it was that order that the employer seeks to appeal from; and it is the violation of that order which the union wants the district judge to punish by proceedings in contempt, a standard remedy for violation of an injunction. If the order to reinstate the employee is an injunction, it is appealable regardless of the fact that the union is seeking additional relief that has not been finally determined by the arbitrator.

The only question therefore is whether the fact that the district court ordered reinstatement in the context of confirming an arbitrator's award takes the case out of the normal application of section 1292(a)(1). We think not. It is true that orders to arbitrate are sui generis. Before the enactment of 9 U.S.C. Sec. 15, which expressly makes interlocutory orders to arbitrate nonappealable, case law denied them the status of mandatory injunctions for the precise purpose of preventing the delay of arbitration through interlocutory appeals. University Life Ins. Co. of America v. Unimarc Ltd., 699 F.2d 846, 849 (7th Cir.1983). But the employer in this case is not trying to appeal from an order to arbitrate, or otherwise to delay the completion of the arbitration. The main part of the arbitration is over; the minor part that remains will continue regardless of the appeal.

So the motion to dismiss the appeal must be denied. But since the ground of appealability is not the existence of a final decision but the issuance of an injunction, the district judge has not been divested of jurisdiction over the case. An interlocutory appeal does not divest the district court of jurisdiction, Shevlin v. Schewe, 809 F.2d 447, 450-51 (7th Cir.1987), here, to determine whether the employer violated the injunction--which had not been stayed pending appeal--when it fired the employee the second time. The district court may wish to defer the contempt proceeding, since the appropriateness of imposing a sanction for violation of an injunction is related to, although not necessarily determined by, the injunction's validity, the question presented by the employer's appeal. But that is a matter for the district court to decide. We have jurisdiction of the employer's appeal, and the motion to dismiss is therefore

DENIED.