348:
282:
Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a
Boolean ring with an identity. The existence of the identity is necessary to consider the ring as an algebra over the
1040:
Unification in
Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise (i.e. if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a
508:
If a
Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra.
1331:
Kandri-Rody, Abdelilah; Kapur, Deepak; Narendran, Paliath (1985). "An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras".
287:: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in
1077:
When a
Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and
332:
531:
of a
Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
1046:
100:). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra,
1394:
1348:
1307:
1229:
1367:
459:. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:
56:
1425:
1244:
1271:
983:, that is, algorithms exist to solve arbitrary equations over Boolean rings. Both unification and matching in
1435:
362:
in a
Boolean algebra is often written additively, it makes sense in this context to denote ring addition by
327:, again with symmetric difference and intersection as operations. More generally with these operations any
1430:
519:
it is a homomorphism of the corresponding
Boolean algebras. Furthermore, a subset of a Boolean ring is a
456:
67:
60:
996:
984:
112:
There are at least four different and incompatible systems of notation for
Boolean rings and algebras:
314:
929:. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.
1469:
961:
1316:
824:
1449:
1464:
1324:
1058:
1042:
980:
310:
86:
1377:
Martin, U.; Nipkow, T. (1986). "Unification in
Boolean Rings". In Jörg H. Siekmann (ed.).
8:
976:
912:
751:
747:
117:
90:
71:
762:
with two elements, in precisely one way. In particular, any finite
Boolean ring has as
455:
These operations then satisfy all of the axioms for meets, joins, and complements in a
177:
75:
30:
1257:
1413:
1390:
1363:
1344:
1303:
1225:
513:
1382:
1336:
1280:
1253:
1217:
1409:
1239:
933:
908:
780:
1242:; Schmidt-Schauß, M. (1989). "Unification of Boolean Rings and Abelian Groups".
1299:
1213:
863:
516:
288:
1386:
960:. Furthermore, as all elements are idempotents, Boolean rings are commutative
1458:
1340:
1078:
895:
891:
528:
336:
328:
1285:
1266:
1138:
767:
367:
351:
101:
82:
848:
is a Boolean ring, since every element in the localization is idempotent.
1191:
1145:
988:
965:
876:
763:
654:
524:
17:
520:
318:
218:
1335:. Lecture Notes in Computer Science. Vol. 202. pp. 345–364.
964:
and hence absolutely flat, which means that every module over them is
354:
for the Boolean operations of conjunction, disjunction, and complement
527:(prime order ideal, maximal order ideal) of the Boolean algebra. The
300:
1180:
1157:
200:
for the join, given in terms of ring notation (given just above) by
872:
is a Boolean ring, since every partial endomorphism is idempotent.
97:
992:
817:
1417:
523:(prime ring ideal, maximal ring ideal) if and only if it is an
347:
1237:
1197:
284:
1107:
1018:
in a Boolean ring can be rewritten as the matching problem
1095:
317:. As another example, we can also consider the set of all
1330:
1186:
274:
for the ring sum, in an effort to avoid the ambiguity of
1169:
999:
Boolean rings. (In fact, as any unification problem
932:
Every finitely generated ideal of a Boolean ring is
911:
and also a Boolean ring, so it is isomorphic to the
1381:. LNCS. Vol. 230. Springer. pp. 506–513.
653:. A similar proof shows that every Boolean ring is
1119:
1456:
190:for the meet (same as the ring product) and use
1212:
342:
1264:
1163:
1139:"Disjunction as sum operation in Boolean Ring"
642:from both sides of this equation, which gives
534:
779:is a Boolean ring: consider for instance the
1376:
1267:"On the ring of quotients of a Boolean ring"
1175:
770:. Not every unital associative algebra over
70:, with ring multiplication corresponding to
1198:Boudet, Jouannaud & Schmidt-Schauß 1989
339:(treated as a ring with these operations).
1357:
1151:
1423:
1284:
255:is different from the use in ring theory.
1315:
1293:
1113:
1101:
1081:is that they have complement operations.
366:, a symbol that is often used to denote
346:
55:, that is, a ring that consists of only
1187:Kandri-Rody, Kapur & Narendran 1985
816:is again a Boolean ring. Likewise, any
1457:
335:every Boolean ring is isomorphic to a
1403:
1333:Rewriting Techniques and Applications
1125:
820:of a Boolean ring is a Boolean ring.
632:is an abelian group, we can subtract
512:A map between two Boolean rings is a
299:One example of a Boolean ring is the
1360:Handbook of Boolean algebras, vol. 1
309:, where the addition in the ring is
1222:Introduction to Commutative Algebra
221:and logic it is also common to use
66:Every Boolean ring gives rise to a
13:
1296:A First Course In Abstract Algebra
746:shows that any Boolean ring is an
14:
1481:
1442:
1265:Brainerd, B.; Lambek, J. (1959).
731:(using the first property above).
1047:minimal complete set of unifiers
1037:, the problems are equivalent.)
923:, which shows the maximality of
120:the standard notation is to use
1245:Journal of Symbolic Computation
1406:Introduction To Modern Algebra
1272:Canadian Mathematical Bulletin
1131:
1071:
971:
851:The maximal ring of quotients
333:Stone's representation theorem
180:, a common notation is to use
1:
1258:10.1016/s0747-7171(89)80054-9
1206:
1362:. Amsterdam: North-Holland.
1088:
343:Relation to Boolean algebras
313:, and the multiplication is
258:A rare convention is to use
59:. An example is the ring of
7:
1431:Encyclopedia of Mathematics
1424:Ryabukhin, Yu. M. (2001) ,
1358:Koppelberg, Sabine (1989).
1052:
862:(in the sense of Utumi and
535:Properties of Boolean rings
294:
107:
96:, which would constitute a
10:
1486:
1294:Fraleigh, John B. (1976),
1164:Brainerd & Lambek 1959
1387:10.1007/3-540-16780-3_115
962:von Neumann regular rings
358:Since the join operation
1341:10.1007/3-540-15976-2_17
1176:Martin & Nipkow 1986
1064:
1404:McCoy, Neal H. (1968),
1214:Atiyah, Michael Francis
987:free Boolean rings are
321:or cofinite subsets of
81:, and ring addition to
1286:10.4153/CMB-1959-006-x
1154:, Definition 1.1, p. 7
355:
331:is a Boolean ring. By
1325:John Wiley & Sons
373:Given a Boolean ring
350:
285:field of two elements
83:exclusive disjunction
1408:(Revised ed.),
1059:Ring sum normal form
1045:, and otherwise the
1043:most general unifier
979:in Boolean rings is
866:) of a Boolean ring
804:of any Boolean ring
311:symmetric difference
264:for the product and
147:for the ring sum of
87:symmetric difference
1116:, pp. 130, 268
748:associative algebra
539:Every Boolean ring
118:commutative algebra
57:idempotent elements
1224:, Westview Press,
1104:, pp. 25, 200
997:finitely presented
985:finitely generated
884:in a Boolean ring
832:of a Boolean ring
794:The quotient ring
568:, because we know
356:
231:for the meet, and
173:for their product.
1396:978-3-540-16780-8
1350:978-3-540-15976-6
1321:Topics In Algebra
1309:978-0-201-01984-1
1231:978-0-201-40751-8
810:modulo any ideal
514:ring homomorphism
61:integers modulo 2
1477:
1448:John Armstrong,
1438:
1420:
1400:
1373:
1354:
1327:
1323:(2nd ed.),
1312:
1298:(2nd ed.),
1290:
1288:
1261:
1240:Jouannaud, J.-P.
1234:
1218:Macdonald, I. G.
1200:
1195:
1189:
1184:
1178:
1173:
1167:
1161:
1155:
1149:
1143:
1142:
1135:
1129:
1123:
1117:
1111:
1105:
1099:
1082:
1075:
1036:
1017:
959:
928:
922:
906:
889:
883:
871:
861:
847:
837:
831:
815:
809:
803:
790:
778:
761:
745:
730:
720:
710:and this yields
709:
652:
641:
631:
620:
567:
561:
555:
544:
504:
476:
450:
437:
413:
396:
390:
384:
378:
365:
361:
326:
308:
277:
273:
263:
254:
250:
240:
230:
213:
199:
189:
172:
158:
152:
146:
95:
80:
54:
48:
42:
28:
1485:
1484:
1480:
1479:
1478:
1476:
1475:
1474:
1470:Boolean algebra
1455:
1454:
1445:
1410:Allyn and Bacon
1397:
1370:
1351:
1317:Herstein, I. N.
1310:
1232:
1209:
1204:
1203:
1196:
1192:
1185:
1181:
1174:
1170:
1162:
1158:
1152:Koppelberg 1989
1150:
1146:
1137:
1136:
1132:
1124:
1120:
1112:
1108:
1100:
1096:
1091:
1086:
1085:
1076:
1072:
1067:
1055:
1019:
1000:
991:, and both are
974:
937:
924:
921:
915:
909:integral domain
898:
885:
879:
867:
852:
839:
833:
827:
811:
805:
795:
789:
783:
781:polynomial ring
777:
771:
760:
754:
736:
722:
711:
661:
643:
633:
625:
572:
563:
557:
546:
540:
537:
479:
463:
457:Boolean algebra
441:
417:
401:
392:
386:
380:
374:
363:
359:
345:
322:
304:
297:
275:
265:
259:
252:
251:. This use of
242:
232:
222:
201:
191:
181:
160:
154:
148:
121:
110:
93:
78:
68:Boolean algebra
50:
44:
34:
24:
12:
11:
5:
1483:
1473:
1472:
1467:
1453:
1452:
1444:
1443:External links
1441:
1440:
1439:
1426:"Boolean ring"
1421:
1401:
1395:
1379:Proc. 8th CADE
1374:
1368:
1355:
1349:
1328:
1313:
1308:
1300:Addison-Wesley
1291:
1262:
1252:(5): 449–477.
1235:
1230:
1208:
1205:
1202:
1201:
1190:
1179:
1168:
1156:
1144:
1130:
1118:
1106:
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1092:
1090:
1087:
1084:
1083:
1069:
1068:
1066:
1063:
1062:
1061:
1054:
1051:
973:
970:
919:
787:
775:
758:
733:
732:
721:, which means
622:
621:
536:
533:
517:if and only if
506:
505:
477:
453:
452:
439:
415:
397:we can define
344:
341:
296:
293:
289:measure theory
280:
279:
256:
215:
174:
109:
106:
9:
6:
4:
3:
2:
1482:
1471:
1468:
1466:
1463:
1462:
1460:
1451:
1450:Boolean Rings
1447:
1446:
1437:
1433:
1432:
1427:
1422:
1419:
1415:
1411:
1407:
1402:
1398:
1392:
1388:
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1375:
1371:
1369:0-444-70261-X
1365:
1361:
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1215:
1211:
1210:
1199:
1194:
1188:
1183:
1177:
1172:
1166:, Corollary 2
1165:
1160:
1153:
1148:
1140:
1134:
1127:
1122:
1115:
1114:Herstein 1975
1110:
1103:
1102:Fraleigh 1976
1098:
1094:
1080:
1079:sigma-algebra
1074:
1070:
1060:
1057:
1056:
1050:
1048:
1044:
1038:
1034:
1030:
1026:
1022:
1015:
1011:
1007:
1003:
998:
994:
990:
986:
982:
978:
969:
967:
963:
957:
953:
949:
945:
941:
935:
930:
927:
918:
914:
910:
905:
901:
897:
896:quotient ring
893:
888:
882:
878:
873:
870:
865:
859:
855:
849:
846:
842:
836:
830:
826:
821:
819:
814:
808:
802:
798:
792:
786:
782:
774:
769:
765:
757:
753:
749:
743:
739:
735:The property
729:
725:
718:
714:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
660:
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629:
619:
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611:
607:
603:
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587:
583:
579:
575:
571:
570:
569:
566:
560:
553:
549:
543:
532:
530:
529:quotient ring
526:
522:
518:
515:
510:
502:
498:
494:
490:
486:
482:
478:
474:
470:
466:
462:
461:
460:
458:
449:
445:
440:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
399:
398:
395:
389:
383:
377:
371:
369:
353:
352:Venn diagrams
349:
340:
338:
337:field of sets
334:
330:
329:field of sets
325:
320:
316:
312:
307:
302:
292:
290:
286:
272:
268:
262:
257:
249:
245:
241:for the join
239:
235:
229:
225:
220:
216:
212:
208:
204:
198:
194:
188:
184:
179:
175:
171:
167:
163:
157:
151:
144:
140:
136:
132:
128:
124:
119:
115:
114:
113:
105:
103:
99:
92:
88:
84:
77:
73:
69:
64:
62:
58:
53:
47:
41:
37:
32:
27:
23:
19:
1429:
1405:
1378:
1359:
1332:
1320:
1295:
1276:
1270:
1249:
1243:
1238:Boudet, A.;
1221:
1193:
1182:
1171:
1159:
1147:
1133:
1128:, p. 46
1121:
1109:
1097:
1073:
1049:is finite).
1039:
1032:
1028:
1024:
1020:
1013:
1009:
1005:
1001:
975:
955:
951:
947:
943:
939:
931:
925:
916:
903:
899:
886:
880:
874:
868:
857:
853:
850:
844:
840:
834:
828:
825:localization
822:
812:
806:
800:
796:
793:
784:
772:
768:power of two
755:
741:
737:
734:
727:
723:
716:
712:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
648:
644:
638:
634:
627:
623:
617:
613:
609:
605:
601:
597:
593:
589:
585:
581:
577:
573:
564:
558:
551:
547:
541:
538:
511:
507:
500:
496:
492:
488:
484:
480:
472:
468:
464:
454:
447:
443:
434:
430:
426:
422:
418:
410:
406:
402:
393:
387:
381:
375:
372:
368:exclusive or
357:
323:
315:intersection
305:
298:
281:
270:
266:
260:
247:
243:
237:
233:
227:
223:
210:
206:
202:
196:
192:
186:
182:
169:
165:
161:
155:
149:
142:
138:
134:
130:
126:
122:
111:
102:George Boole
65:
51:
45:
39:
35:
25:
22:Boolean ring
21:
15:
1465:Ring theory
989:NP-complete
977:Unification
972:Unification
877:prime ideal
764:cardinality
655:commutative
525:order ideal
303:of any set
91:disjunction
72:conjunction
18:mathematics
1459:Categories
1207:References
1126:McCoy 1968
624:and since
545:satisfies
521:ring ideal
219:set theory
159:, and use
33:for which
1436:EMS Press
1279:: 25–29.
1089:Citations
981:decidable
936:(indeed,
934:principal
838:by a set
750:over the
301:power set
1418:68015225
1319:(1975),
1220:(1969),
1053:See also
556:for all
295:Examples
108:Notation
98:semiring
43:for all
993:NP-hard
892:maximal
818:subring
137:) ∨ (¬
1416:
1393:
1366:
1347:
1306:
1228:
907:is an
894:: the
875:Every
864:Lambek
495:) ∧ ¬(
446:= 1 ⊕
379:, for
319:finite
1065:Notes
1035:) = 0
946:) = (
913:field
752:field
178:logic
89:(not
29:is a
1414:LCCN
1391:ISBN
1364:ISBN
1345:ISBN
1304:ISBN
1226:ISBN
1027:) +
1008:) =
966:flat
823:Any
677:) =
630:, ⊕)
588:) =
385:and
153:and
133:∧ ¬
76:meet
31:ring
20:, a
1383:doi
1337:doi
1281:doi
1254:doi
995:in
890:is
744:= 0
719:= 0
669:= (
651:= 0
580:= (
562:in
554:= 0
487:= (
391:in
291:.)
217:In
176:In
129:= (
116:In
85:or
74:or
49:in
16:In
1461::
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1428:,
1412:,
1389:.
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1275:.
1269:.
1248:.
1216:;
968:.
958:))
956:xy
954:+
950:+
902:/
843:⊆
829:RS
799:/
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766:a
740:⊕
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726:=
724:xy
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713:xy
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689:⊕
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637:⊕
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608:⊕
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600:⊕
596:⊕
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499:∧
491:∨
483:⊕
471:∧
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465:xy
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405:∧
370:.
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162:xy
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1399:.
1385::
1372:.
1353:.
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1029:g
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887:R
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759:2
756:F
742:x
738:x
707:y
695:x
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364:⊕
360:∨
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94:∨
79:∧
52:R
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26:R
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