208 F.3d 610 (7th Cir. 2000)
Jays Foods, L.L.C.,    Plaintiff-Appellee,v.Chemical & Allied Product Workers Union,  Local 20, AFL-CIO,    Defendant-Appellant.
No. 99-2807
In the  United States Court of Appeals  For the Seventh Circuit
Argued January 21, 2000Decided March 31, 2000Rehearing and Rehearing En BancDenied April 28, 2000

Appeal from the United States District Court  for the Northern District of Illinois, Eastern Division.  No. 97 C 6227--Wayne R. Andersen, Judge. [Copyrighted Material Omitted]
Before Posner, Chief Judge, and Flaum and Diane P.  Wood, Circuit Judges.
Posner, Chief Judge.


1
We write to untie a  procedural knot partly of our own tying. Jays  Food and a local of the chemical workers union  have a collective bargaining agreement that  provides for arbitration of disputes arising  under it. In 1997 an arbitrator determined that  Jays had violated the agreement by starting a  third shift in a manner that reduced the paid  hours of work of employees on the first and  second shifts. Jays filed suit in federal  district court to vacate the arbitrator's  decision, and the union counterclaimed to confirm  it. 9 U.S.C. sec. 9; 29 U.S.C. sec. 185. The  district court ruled that the arbitrator had  exceeded his authority under the collective  bargaining agreement, and remanded the matter to  the arbitrator "for further determination,"  adding that by doing this the court was  "terminating case." After the arbitrator rendered  his decision on remand, this time in favor of  Jays, the union appealed from the order of  remand. We dismissed the appeal in an unpublished  order, on the ground that an order by a district  court remanding a case to an arbitrator is not a  final, appealable decision within the meaning of  28 U.S.C. sec. 1291 unless the remand requires a  merely ministerial act, such as calculating  interest when the period for which the  calculation must be made, the interest rate, the  principal to which the rate is to be applied, and  whether interest is simple or compound have all  been determined. Shearson Loeb Rhoades, Inc. v.  Much, 754 F.2d 773, 776-78 (7th Cir. 1985);  United Steelworkers v. Aurora Equipment Co., 830  F.2d 753, 754 (7th Cir. 1987). We had pointed out  in these cases that if more is required, allowing  an appeal would delay the ultimate resolution of  the parties' dispute. More was required here. The  arbitrator had written an opinion that did not  satisfy the district judge, but the judge sent it  back to him to make a new award rather than  telling him to rule in favor of Jays.


2
It is true that by saying "terminating case" the  district court might have been thought to be  issuing a final judgment; but all he might have  meant was that he was terminating the immediate  proceeding before him, that is, the motion to  vacate the arbitrator's award and the  countermotion to confirm it. And whatever he  meant, that may be all he did. For as we said in  Chrysler Motors Corp. v. International Union,  Allied Industrial Workers of America, 909 F.2d  248, 249 (7th Cir. 1990) (emphasis added), "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, . . . 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."


3
But we were wrong, not in general but with  reference to the appealability of orders  remanding cases to arbitrators, in thinking that  the district court's order of remand was not  immediately appealable. Congress had in 1988  amended the Federal Arbitration Act to make  orders vacating arbitral awards appealable  immediately, though nonfinal, 9 U.S.C. sec.  16(a)(1)(E); Perlman v. Swiss Bank Corp.  Comprehensive Disability Protection Plan, 195  F.3d 975, 979-80 (7th Cir. 1999), unless the  purpose of the remand was merely to enable the  arbitrator to clarify his decision in order to  set the stage for informed appellate review.  Virgin Islands Housing Authority v. Coastal  General Construction Services Corp., 27 F.3d 911,  913-14 (3d Cir. 1994). That was not the purpose  here. Our error was harmless, however, because  the appeal was in any event untimely. The  district court had issued its order of remand on  May 11, 1998, and the union did not file its  notice of appeal until August 19, which was more  than 30 days later.


4
We ended our order dismissing the union's appeal  by saying that "the parties must proceed back  through the federal courts to confirm or vacate  the second award." This Delphic utterance left  unexplained both how the union could challenge  the district court's first decision, the decision  vacating the arbitrator's award, which had been  in the union's favor, and why the parties would  have to file a brand-new suit (if that's what  "proceed back" meant) in the district court.  Let's take the second question first. There is no  hard-and-fast rule, except in the Ninth Circuit,  Shapiro v. Paradise Valley Unified School Dist.  No. 69, 152 F.3d 1159 (9th Cir. 1998) (per  curiam), and, by virtue of statute, in the social  security area, 42 U.S.C. sec.sec. 405(g); Shalala  v. Schaefer, 509 U.S. 292 (1993); Richmond v.  Chater, 94 F.3d 263, 266-70 (7th Cir. 1996);  Raitport v. Callahan, 183 F.3d 101 (2d Cir. 1999)  (per curiam); cf. Perlman v. Swiss Bank Corp.  Comprehensive Disability Protection Plan, supra,  195 F.3d at 979-80, and perhaps other specialized  areas with which we are not familiar, about when  a remanding court loses jurisdiction over the  case, thus necessitating a fresh proceeding of  some sort after the remand to bring the issue  back to that court. The presumption, however, is  in favor of relinquishment. For when a court  makes a limited remand unlikely to resolve the  case but intended rather to assist the court in  making its decision (a typical example being a  remand to enable the district court to decide  whether to enter a Rule 54(b) order to enable us  to exercise appellate jurisdiction), the court  will either state that it is retaining  jurisdiction, e.g., Boyko v. Anderson, 185 F.3d  672 (7th Cir. 1999); Alexander v. Chicago Park  District, 773 F.2d 850, 857 (7th Cir. 1985);  United States v. Doe, 63 F.3d 121, 131 (2d Cir.  1995); Caterpillar, Inc. v. N.L.R.B., 138 F.3d  1105, 1107-08 (7th Cir. 1998) (chambers opinion),  and cases cited there, or dismiss with leave to  reinstate, which we treat as the equivalent of a  stay. Baltimore & Ohio Chicago Terminal R.R. v.  Wisconsin Central Limited, 154 F.3d 404, 408 (7th  Cir. 1998). The presumption was reinforced here  by the district court's statement that it was  "terminating case," suggesting that it was indeed  relinquishing jurisdiction. No doubt the judge  hoped that on remand the arbitrator would render  an award that satisfied both parties or, more  likely, would be solid enough to discourage a  judicial challenge.


5
On remand, the arbitrator grudgingly ruled in  favor of Jays, emphasizing that he disagreed with  the district court's decision but was bowing to  what he thought its implicit command to rule for  the company. Naturally at this point the union  wanted to get appellate review of the district  court's initial decision, the decision vacating  the arbitral award in the union's favor. It could  have gotten this by filing a proceeding in the  district court to set aside the second arbitral  award on the ground that the court had erred in  setting aside the first one. If the court was not  persuaded and ruled in favor of Jays, the union  could appeal to us and the appeal would bring up  for review all nonmoot interlocutory orders  issued by the district court, Lauer v. Apfel, 169  F.3d 489, 492 (7th Cir. 1999); LeBlang Motors,  Ltd. v. Subaru of America, Inc., 148 F.3d 680,  689 (7th Cir. 1998); Librizzi v. Children's  Memorial Medical Center, 134 F.3d 1302, 1305-06  (7th Cir. 1998); Keefe v. Prudential Property &  Casualty Ins. Co., 203 F.3d 218, 226 (3d Cir.  2000), including the order vacating the first  arbitral award.


6
Granted, this is a slightly tricky point, since  in our hypothetical example of the union's filing  a fresh proceeding to set aside the award made by  the arbitrator on remand we described as  "interlocutory" an order technically issued in a  prior case. But the usage is correct. When a case  is appealed (here to the district court from the  arbitrator rather than from the district court to  this court) following a remand, there are two  appellate proceedings but only one underlying  litigation, and any interlocutory order in that  litigation, even if it preceded the first appeal,  is (within the limits of the doctrine of law of  the case) open to review on the second appeal,  unless it has become moot in the interim.  Shearson Loeb Rhoades, Inc. v. Much, supra; cf.  Reilly v. Waukesha County, 993 F.2d 1284, 1286-87  (7th Cir. 1993). The fact that an interlocutory  order is appealable doesn't require that it be  appealed, on pain of forfeiture if it is not.  That would only encourage piecemeal litigation. A  party can wait until the litigation is over and  then bring a single appeal from the final  judgment and challenge all nonmoot interlocutory  orders, appealable or not, rendered along the  way. Retired Chicago Police Ass'n v. City of  Chicago, 7 F.3d 584, 608 (7th Cir. 1993);  Chambers v. Ohio Dept. of Human Services, 145  F.3d 793, 796 (6th Cir. 1998); Ernst v. Child &  Youth Services, 108 F.3d 486, 492-93 (3d Cir.  1997); Sierra Club v. Robertson, 28 F.3d 753, 756  n. 3 (11th Cir. 1994).


7
Instead of taking up our invitation to file a  new suit to challenge the second arbitral award,  the union inexplicably moved the court in the  original suit, the suit the district court had  dismissed when it remanded the case to the  arbitrator--a suit that consequently did not  exist when the union filed its motion--to enter a  "Rule 54" order. It is from the denial of that  motion that the union appeals. The reference in  the motion to "Rule 54" was presumably to Rule  54(b) of the Federal Rules of Civil Procedure,  which permits a district court in certain  circumstances to issue a final, and thus  immediately appealable, judgment even though some  piece of the case remains pending in the district  court. E.g., Continental Casualty Co. v. Anderson  Excavating & Wrecking Co., 189 F.3d 512, 516-18  (7th Cir. 1999). The union must have thought it  couldn't appeal the district court's order  vacating the initial arbitral award without  getting the court to make that order a Rule 54(b)  judgment, because the order was interlocutory  (and thus unappealable unless made a Rule 54(b)  judgment); it must have thought the case was  continuing in the district court awaiting a  motion to vacate or confirm the second award.  This was doubly wrong. The order vacating the  arbitrator's first award, though indeed  interlocutory, could, as we have emphasized, be  appealed at the end of the entire litigation. And  there was no pending proceeding in the district  court, which had relinquished jurisdiction over  the controversy when it sent the case back to the  arbitrator. It is not surprising, therefore, that  the district court's response to the Rule 54(b)  motion was to deny it, on the ground that there  was no case before the court. The motion was  frivolous, and in no wise invited by our previous  order. What is inexplicable, therefore, is why  the union did not file a fresh challenge to the  arbitrator's second award, which if confirmed by  the district court could have been brought up to  us for review by an appeal from that  confirmation.


8
The denial of the union's motion for the entry  of a Rule 54(b) judgment was correct, and is  therefore


9
Affirmed.

