3424:
1223:
384:. As noted in the previous example, the inversion map is not necessarily the only map with this property in an inverse semigroup. There may well be other involutions that leave all idempotents invariant; for example the identity map on a commutative regular, hence inverse, semigroup, in particular, an abelian group. A
1047:
The first of these looks like the definition of a regular element, but is actually in terms of the involution. Likewise, the second axiom appears to be describing the commutation of two idempotents. It is known however that regular semigroups do not form a variety because their class does not contain
1056:
in 1968). This line of reasoning motivated
Nordahl and Scheiblich to begin in 1977 the study of the (variety of) *-semigroups that satisfy only the first these two axioms; because of the similarity in form with the property defining regular semigroups, they named this variety regular *-semigroups.
1207:
the entire semilattice of idempotents is a p-system. Also, if a regular semigroup S has a p-system that is multiplicatively closed (i.e. subsemigroup), then S is an inverse semigroup. Thus, a p-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup.
1868:
1072:
is a regular *-semigroup that is not an inverse semigroup. It is also easy to verify that in a regular *-semigroup the product of any two projections is an idempotent. In the aforementioned rectangular band example, the projections are elements of the form
2292:
1202:
A regular semigroup S is a *-regular semigroup, as defined by
Nordahl & Scheiblich, if and only if it has a p-system F(S). In this case F(S) is the set of projections of S with respect to the operation ° defined by F(S). In an
2899:
1751:
2996:
3360:
3254:
2938:
2641:
2159:
2028:
1657:
2549:
978:
There are two related, but not identical notions of regularity in *-semigroups. They were introduced nearly simultaneously by
Nordahl & Scheiblich (1978) and respectively Drazin (1979).
3108:
3054:
2373:
965:
1578:
2831:
994:
are a subclass of *-semigroups. It is also textbook knowledge that an inverse semigroup can be characterized as a regular semigroup in which any two idempotents commute. In 1963,
3300:
3185:. It can be shown than the order of rewriting (deleting) such pairs does not matter, i.e. any order of deletions produces the same result. (Otherwise put, these rules define a
1715:
3389:
2400:
2718:
2573:
2195:
3411:. (This latter choice of terminology conflicts however with the use of "involutive" to denote any semigroup with involution—a practice also encountered in the literature.)
3183:
3153:
78:
It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.
2435:
1442:
1416:
1136:
The problem of characterizing when a regular semigroup is a regular *-semigroup (in the sense of
Nordahl & Scheiblich) was addressed by M. Yamada (1982). He defined a
2064:
2669:
1599:
2087:
2774:
2746:
2608:
2494:
2455:
2327:
2187:
1952:
1899:
1735:
1677:
3264:
to multiple kinds of "parentheses" However simplification in the Dyck congruence takes place regardless of order. For example, if ")" is the inverse of "(", then
2799:
2690:
1924:
2107:
1494:) is a *-regular semigroup with this involution. The Moore–Penrose inverse of A in this *-regular semigroup is the classical Moore–Penrose inverse of
3395:. The quotient of a free monoid with involution by the Shamir congruence is not a group, but a monoid ; nevertheless it has been called the
2839:
1081:) and are idempotent. However, two different projections in this band need not commute, nor is their product necessarily a projection since (
3994:
3966:
3934:
3909:
3868:
3820:
3682:
3644:
3619:
1863:{\displaystyle y^{\dagger }={\begin{cases}\theta (y)&{\text{if }}y\in X\\\theta ^{-1}(y)&{\text{if }}y\in X^{\dagger }\end{cases}}}
801:
In terms of examples for these notions, in the *-semigroup of binary relations on a set, the partial isometries are the relations that are
2465:
is a distinct element in an alphabet with involution, and consequently the same observation extends to a free semigroup with involution.
3718:
Easdown, David, and W. D. Munn. "On semigroups with involution." Bulletin of the
Australian Mathematical Society 48.01 (1993): 93–100.
4061:
2943:
3186:
2457:
in this choice of terminology is explained below in terms of the universal property of the construction.) Note that unlike in
4090:
3454:
A Baer *-semigroup is a *-semigroup with (two-sided) zero in which the right annihilator of every element coincides with the
3305:
3199:
311:
2904:
314:
remarked that it "becomes clear when we think of and as the operations of putting on our socks and shoes, respectively."
4153:
3590:
3117:
is not very far off from that of a free monoid with involution. The additional ingredient needed is to define a notion of
2613:
2112:
1394:
One motivation for studying these semigroups is that they allow generalizing the Moore–Penrose inverse's properties from
3190:
806:
1964:
1623:
366:
both this map and the one from the previous example are involutions satisfying the axioms of semigroup with involution.
4083:
Semigroup theory and its applications: proceedings of the 1994 conference commemorating the work of Alfred H. Clifford
2502:
1611:
999:
155:
1388:
575:
are always hermitian, and so are all powers of a hermitian element. As noted in the examples section, a semigroup
4158:
464:
163:
3067:
3013:
2332:
3003:
1124:(but not necessarily the antidistributivity of * over multiplication) have also been studied under the name of
457:
183:
66:
48:
3189:
rewriting system.) Equivalently, a free group is constructed from a free monoid with involution by taking the
912:
1544:
1526:
381:
2834:
2804:
1680:
594:
Certain basic concepts may be defined on *-semigroups in a way that parallels the notions stemming from a
3900:(2009). "Algebraic Systems and Pushdown Automata". In Manfred Droste; Werner Kuich; Heiko Vogler (eds.).
3267:
1450:
877:
86:
2287:{\displaystyle w^{\dagger }=w_{k}^{\dagger }w_{k-1}^{\dagger }\cdots w_{2}^{\dagger }w_{1}^{\dagger }.}
1686:
1060:
It is a simple calculation to establish that a regular *-semigroup is also a regular semigroup because
3563:
998:
showed that the following two axioms provide an analogous characterization of inverse semigroups as a
762:
3567:
3365:
3007:
2378:
1293:
1286:
1053:
424:
131:
97:
2695:
2557:
1773:
1292:. This defining property can be formulated in several equivalent ways. Another is to say that every
3194:
3158:
3128:
3061:
1475:
846:
434:
343:
55:
2780:. In the case of the free semigroup with involution, given an arbitrary semigroup with involution
2413:
1425:
1399:
4134:
4032:
Foulis, D. J. Relative inverses in Baer *-semigroups. Michigan Math. J. 10 (1963), no. 1, 65–84.
3510:
2033:
105:
2653:
1583:
4086:
4057:
3990:
3962:
3930:
3905:
3864:
3816:
3678:
3640:
3615:
3529:
2777:
2776:). The qualifier "free" for these constructions is justified in the usual sense that they are
2072:
1372:
1204:
991:
595:
588:
580:
531:
453:
389:
385:
374:
332:
179:
109:
51:
2751:
2723:
2585:
2471:
2440:
2304:
2164:
1929:
1876:
1720:
1662:
1140:
F(S) as subset of the idempotents of S, denoted as usual by E(S). Using the usual notation V(
62:, exhibits certain fundamental properties of the operation of taking the inverse in a group:
4054:
4033:
4014:
3556:
3548:
2576:
1738:
813:
766:
552:
475:
307:
32:
471:
under the operation of concatenation of sequences, with sequence reversal as an involution.
130:, which has the same form of interaction with multiplication as taking inverses has in the
17:
4115:
4104:
3579:
3455:
3125:
rule for producing such words simply by deleting any adjacent pairs of letter of the form
1507:
995:
445:
218:
175:
143:
139:
2783:
2674:
1908:
1514:. The construction of a free semigroup (or monoid) with involution is based on that of a
116:
is an involution because the transpose is well defined for any matrix and obeys the law
2092:
1902:
1602:
1515:
429:
147:
82:
74:
The same interaction law with the binary operation as in the case of the group inverse.
59:
3813:
The Theory of 2-structures: A Framework for
Decomposition and Transformation of Graphs
3423:
1222:
193:(in Russian) as result of his attempt to bridge the theory of semigroups with that of
4147:
4122:
3544:
3261:
2999:
1955:
1534:
1522:
can easily be derived by refining the construction of a free monoid with involution.
694:
363:
299:
Semigroups that satisfy only the first of these axioms belong to the larger class of
190:
159:
3612:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
3897:
3118:
1530:
802:
392:
if and only if it admits an involution under which each idempotent is an invariant.
167:
151:
134:(which is a subgroup of the full linear monoid). However, for an arbitrary matrix,
3675:
Harmonic
Analysis on Semigroups: Theory of Positive Definite and Related Functions
3533:
3057:
2497:
1511:
1125:
1049:
893:
468:
328:
300:
93:
28:
4125:", Bulletin of the Society of Mathematicians of Banja Luka Vol. 9 (2002), 7–47.
4053:
Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries".
1383:′ satisfying these axioms is unique. It is called the Moore–Penrose inverse of
1299:
contains a projection. An axiomatic definition is the condition that for every
4130:
4018:
3400:
3114:
3060:. An analogous argument holds for the free monoid with involution in terms of
2552:
1519:
400:
396:
378:
171:
4071:
4037:
3585:
3122:
1742:
1538:
210:
113:
44:
4009:
3555:
is a Baer *-semigroup. The involution in this case maps an operator to its
2894:{\displaystyle {\overline {\Phi }}:(X\sqcup X^{\dagger })^{+}\rightarrow S}
514:), with the involution being the order reversal of the elements of a pair (
4129:
This article incorporates material from Free semigroup with involution on
1529:
of a free semigroup with involution are the elements of the union of two (
1518:(and respectively that of a free monoid). Moreover, the construction of a
816:
in a C*-algebra are exactly those defined in this section. In the case of
1183:
is in F(S), where ° is the well-defined operation from the previous axiom
650:. Every projection is a partial isometry, and for every partial isometry
194:
4093:. This is a recent survey article on semigroup with (special) involution
189:
Semigroups with involution appeared explicitly named in a 1953 paper of
460:. This is an example of a *-semigroup which is not a regular semigroup.
2650:
The construction above is actually the only way to extend a given map
4123:
Varieties of involution semigroups and involution semirings: a survey
2580:
970:
Another simple example of these notions appears in the next section.
213:
with its binary operation written multiplicatively. An involution in
89:
3458:
of some projection; this property is expressed formally as: for all
3302:; the one-sided congruence that appears in the Dyck language proper
1233: with: clarify motivation for studying these. You can help by
591:
and admits an involution such that every idempotent is hermitian.
306:
In some applications, the second of these axioms has been called
162:
as the binary operation, and the involution being the map which
3418:
2991:{\displaystyle \iota :X\rightarrow (X\sqcup X^{\dagger })^{+}}
1217:
296:
with the involution * is called a semigroup with involution.
3952:
3950:
3948:
3946:
3064:
and the uniqueness up to isomorphism of the construction of
3989:. Springer Science & Business Media. pp. 101–102.
1856:
4103:
Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups,
3811:
Andrzej
Ehrenfeucht; T. Harju; Grzegorz Rozenberg (1999).
1926:
in the usual way with the binary (semigroup) operation on
4100:, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
3677:. Springer Science & Business Media. pp. 87–88.
3673:
C. van den Berg; J. P. R. Christensen; P. Ressel (2012).
1605:.) In the case were the two sets are finite, their union
377:
then the inversion map is an involution which leaves the
3705:
3703:
463:
If X is a set, then the set of all finite sequences (or
178:
between a set and itself, with the involution being the
170:
of the letters in a string. A third example, from basic
3806:
3804:
3582:(aka category with involution) — generalizes the notion
3505:
More recently, Baer *-semigroups have been also called
3435:
3355:{\displaystyle \{(xx^{\dagger },\varepsilon ):x\in X\}}
3249:{\displaystyle \{(yy^{\dagger },\varepsilon ):y\in Y\}}
1234:
555:) when it is left invariant by the involution, meaning
3368:
3308:
3270:
3202:
3161:
3131:
3070:
3016:
2946:
2933:{\displaystyle \Phi =\iota \circ {\overline {\Phi }}}
2907:
2842:
2807:
2786:
2754:
2726:
2698:
2677:
2656:
2616:
2588:
2560:
2551:, which is just the free semigroup extended with the
2505:
2474:
2443:
2416:
2381:
2335:
2307:
2198:
2167:
2115:
2095:
2075:
2036:
1967:
1932:
1911:
1879:
1754:
1723:
1689:
1665:
1626:
1586:
1547:
1428:
1402:
915:
3532:, in particular as the multiplicative semigroups of
2636:{\displaystyle \varepsilon ^{\dagger }=\varepsilon }
2154:{\displaystyle {}^{\dagger }:Y^{+}\rightarrow Y^{+}}
3635:Chris Brink; Wolfram Kahl; Gunther Schmidt (1997).
1387:. This agrees with the classical definition of the
547:of a semigroup with involution is sometimes called
4070:, PhD Thesis, Tulane University, New Orleans, LA.
3957:Enrico G. Beltrametti; Gianni Cassinelli (2010) .
3383:
3354:
3294:
3248:
3177:
3147:
3102:
3048:
2990:
2932:
2893:
2825:
2793:
2768:
2740:
2712:
2684:
2663:
2635:
2602:
2567:
2543:
2488:
2449:
2429:
2394:
2367:
2321:
2286:
2181:
2161:defined as the string reversal of the elements of
2153:
2101:
2081:
2058:
2022:
1946:
1918:
1893:
1862:
1729:
1709:
1671:
1651:
1593:
1572:
1436:
1410:
959:
310:. Regarding the natural philosophy of this axiom,
2023:{\displaystyle w=w_{1}w_{2}\cdots w_{k}\in Y^{+}}
1652:{\displaystyle \theta :X\rightarrow X^{\dagger }}
4135:Creative Commons Attribution/Share-Alike License
4081:, in A.H. Clifford, K.H. Hofmann, M.W. Mislove,
2544:{\displaystyle Y^{*}=Y^{+}\cup \{\varepsilon \}}
138:does not equal the identity element (namely the
3854:
3852:
3850:
3843:. Cambridge University Press. pp. 305–306.
3521:The set of all binary relations on a set (from
3403:—although more recently it has been called the
3056:as a semigroup with involution is unique up to
2410:. (The irrelevance of the concrete identity of
982:Regular *-semigroups (Nordahl & Scheiblich)
618:; the set of partial isometries of a semigroup
54:, which—roughly speaking—brings it closer to a
3834:
3832:
3614:. American Mathematical Society. p. 265.
1148:, F(S) needs to satisfy the following axioms:
805:. The projections in this *-semigroup are the
909:) form an inverse semigroup with the product
8:
3980:
3978:
3349:
3309:
3243:
3203:
2538:
2532:
1717:essentially by taking the disjoint union of
456:, and the multiplication given by the usual
182:, and the multiplication given by the usual
4028:
4026:
3961:. Cambridge University Press. p. 178.
3547:, then the multiplicative semigroup of all
2610:), and suitably extend the involution with
71:Double application "cancelling itself out".
3528:Baer *-semigroups are also encountered in
3103:{\displaystyle (X\sqcup X^{\dagger })^{*}}
3049:{\displaystyle (X\sqcup X^{\dagger })^{+}}
2368:{\displaystyle (X\sqcup X^{\dagger })^{+}}
3987:Lattices and Ordered Algebraic Structures
3863:. American Mathematical Soc. p. 86.
3367:
3322:
3307:
3269:
3216:
3201:
3166:
3160:
3139:
3130:
3094:
3084:
3069:
3040:
3030:
3015:
2982:
2972:
2945:
2920:
2906:
2879:
2869:
2843:
2841:
2806:
2790:
2785:
2765:
2759:
2753:
2737:
2731:
2725:
2709:
2703:
2697:
2681:
2676:
2660:
2655:
2621:
2615:
2599:
2593:
2587:
2564:
2559:
2523:
2510:
2504:
2485:
2479:
2473:
2442:
2421:
2415:
2402:is a semigroup with involution, called a
2391:
2385:
2383:
2380:
2359:
2349:
2334:
2318:
2312:
2306:
2275:
2270:
2260:
2255:
2242:
2231:
2221:
2216:
2203:
2197:
2178:
2172:
2166:
2145:
2132:
2119:
2117:
2114:
2094:
2074:
2041:
2035:
2014:
2001:
1988:
1978:
1966:
1943:
1937:
1931:
1915:
1910:
1890:
1884:
1878:
1847:
1832:
1812:
1790:
1768:
1759:
1753:
1722:
1690:
1688:
1664:
1643:
1625:
1590:
1585:
1564:
1546:
1430:
1429:
1427:
1404:
1403:
1401:
1198:is in E(S); note: not necessarily in F(S)
945:
926:
914:
490:, with the semigroup product defined as (
452:is a *-semigroup with the * given by the
4118:, 24(1), December 1982, pp. 173–187
2468:If in the above construction instead of
1601:emphasized that the union is actually a
241:*) satisfying the following conditions:
3882:
3880:
3602:
960:{\displaystyle A(A^{*}A\wedge BB^{*})B}
58:because this involution, considered as
3927:Holomorphy and Convexity in Lie Theory
3794:
3792:
3737:
3735:
3733:
3637:Relational Methods in Computer Science
2298:
2189:that consist of more than one letter:
1573:{\displaystyle Y=X\sqcup X^{\dagger }}
1261:(in the sense of Drazin) if for every
1510:of semigroups with involution admits
634:that is also hermitian, meaning that
482:with itself, i.e. with elements from
362:is an involution. Furthermore, on an
7:
4085:, Cambridge University Press, 1996,
3815:. World Scientific. pp. 13–14.
3391:is (perhaps confusingly) called the
2826:{\displaystyle \Phi :X\rightarrow S}
1156:in S, there exists a unique a° in V(
3786:Nordahl and Scheiblich, Theorem 2.5
3260:—in a certain sense it generalizes
4098:Regular semigroups with involution
3498:is in fact uniquely determined by
3295:{\displaystyle ()=)(=\varepsilon }
2922:
2908:
2845:
2808:
1481:* is an involution. The semigroup
25:
4121:S. Crvenkovic and Igor Dolinka, "
3929:. Walter de Gruyter. p. 21.
1710:{\displaystyle {}\dagger :Y\to Y}
1470:, the map which assigns a matrix
1257:with an involution * is called a
3422:
3256:, which is sometimes called the
2109:is then extended as a bijection
1221:
1064:* turns out to be an inverse of
892:. Since projections form a meet-
478:on a Cartesian product of a set
346:then the inversion map * :
142:). Another example, coming from
4112:P-systems in regular semigroups
3384:{\displaystyle ()=\varepsilon }
2395:{\displaystyle {}^{\dagger }\,}
1277:-equivalent to some inverse of
399:is a *-semigroup. An important
225:(or, a transformation * :
4133:, which is licensed under the
4011:Journal of Philosophical Logic
3959:The Logic of Quantum Mechanics
3372:
3369:
3334:
3312:
3283:
3280:
3274:
3271:
3228:
3206:
3091:
3071:
3037:
3017:
2979:
2959:
2956:
2885:
2876:
2856:
2817:
2713:{\displaystyle X^{\dagger }\,}
2568:{\displaystyle \varepsilon \,}
2404:free semigroup with involution
2356:
2336:
2138:
1827:
1821:
1785:
1779:
1701:
1636:
1502:Free semigroup with involution
1466:) of square matrices of order
951:
919:
596:regular element in a semigroup
1:
3902:Handbook of Weighted Automata
3610:Christopher Hollings (2014).
3591:Special classes of semigroups
3362:, which instantiates only to
3178:{\displaystyle x^{\dagger }x}
3148:{\displaystyle xx^{\dagger }}
3110:as a monoid with involution.
1212:*-regular semigroups (Drazin)
1108:Semigroups that satisfy only
987:
868:, then the unique projection
856:. For any two projection, if
807:partial equivalence relations
539:Basic concepts and properties
3861:Symmetric Inverse Semigroups
2925:
2848:
2430:{\displaystyle X^{\dagger }}
1437:{\displaystyle \mathbb {C} }
1411:{\displaystyle \mathbb {R} }
1068:. The rectangular band from
896:, the partial isometries on
705:defined as holding whenever
530:). This semigroup is also a
112:which sends a matrix to its
4072:Publications of D.J. Foulis
3841:Elements of Automata Theory
3562:Baer *-semigroup allow the
3513:who studied them in depth.
2645:free monoid with involution
2059:{\displaystyle w_{i}\in Y.}
1506:As with all varieties, the
18:Free monoid with involution
4175:
3925:Karl-Hermann Neeb (2000).
3466:there exists a projection
693:Partial isometries can be
622:is usually abbreviated PI(
467:) of members of X forms a
4019:10.1007/s10992-013-9275-5
3904:. Springer. p. 271.
3859:Stephen Lipscomb (1996).
3525:) is a Baer *-semigroup.
3517:Examples and applications
2664:{\displaystyle \theta \,}
1594:{\displaystyle \sqcup \,}
1052:(a result established by
630:is an idempotent element
444:is a set, the set of all
37:semigroup with involution
4074:(Accessed on 5 May 2009)
3661:Introduction to Geometry
3522:
3399:by its first discoverer—
3004:composition of functions
2458:
2082:{\displaystyle \dagger }
1539:bijective correspondence
1375:first proved that given
1307:there exists an element
1069:
458:composition of relations
184:composition of relations
3639:. Springer. p. 4.
2778:universal constructions
2769:{\displaystyle Y^{*}\,}
2741:{\displaystyle Y^{+}\,}
2603:{\displaystyle Y^{*}\,}
2489:{\displaystyle Y^{+}\,}
2450:{\displaystyle \theta }
2322:{\displaystyle Y^{+}\,}
2182:{\displaystyle Y^{+}\,}
1947:{\displaystyle Y^{+}\,}
1894:{\displaystyle Y^{+}\,}
1730:{\displaystyle \theta }
1672:{\displaystyle \theta }
1609:is sometimes called an
829:) more can be said. If
757:. In a *-semigroup, PI(
563:. Elements of the form
4038:10.1307/mmj/1028998825
3777:Crvenkovic and Dolinka
3709:Nordahl and Scheiblich
3385:
3356:
3296:
3250:
3179:
3149:
3113:The construction of a
3104:
3050:
3010:. The construction of
2992:
2934:
2895:
2835:semigroup homomorphism
2827:
2795:
2770:
2742:
2720:, to an involution on
2714:
2686:
2665:
2637:
2604:
2569:
2545:
2490:
2451:
2431:
2396:
2369:
2329:. Thus, the semigroup
2323:
2288:
2183:
2155:
2103:
2083:
2060:
2024:
1948:
1920:
1895:
1864:
1731:
1711:
1673:
1653:
1595:
1574:
1446:to more general sets.
1438:
1412:
1144:) for the inverses of
961:
837:are projections, then
753:* for some projection
674:are projections, then
666:* are projections. If
335:of S is an involution.
4068:Involution Semigroups
3839:Jacques Sakarovitch.
3759:Lawson p.122 and p.35
3568:orthomodular lattices
3386:
3357:
3297:
3251:
3193:of the latter by the
3180:
3150:
3105:
3051:
2993:
2935:
2896:
2828:
2796:
2771:
2743:
2715:
2687:
2666:
2638:
2605:
2570:
2546:
2491:
2452:
2437:and of the bijection
2432:
2397:
2370:
2324:
2289:
2184:
2156:
2104:
2084:
2061:
2025:
1949:
1921:
1896:
1865:
1732:
1712:
1674:
1654:
1596:
1580:. (Here the notation
1575:
1439:
1413:
1389:Moore–Penrose inverse
1179:in S, and b in F(S),
974:Notions of regularity
962:
878:orthogonal complement
4154:Algebraic structures
4107:, 16(1978), 369–377.
3366:
3306:
3268:
3200:
3159:
3129:
3068:
3062:monoid homomorphisms
3014:
2944:
2905:
2840:
2805:
2784:
2752:
2724:
2696:
2675:
2654:
2614:
2586:
2558:
2503:
2472:
2441:
2414:
2379:
2333:
2305:
2196:
2165:
2113:
2093:
2073:
2034:
1965:
1930:
1909:
1877:
1752:
1737:(as a set) with its
1721:
1687:
1663:
1624:
1584:
1545:
1426:
1400:
1391:of a square matrix.
986:As mentioned in the
913:
174:, is the set of all
132:general linear group
4079:Special Involutions
4066:D J Foulis (1958).
3985:T.S. Blyth (2006).
2794:{\displaystyle S\,}
2685:{\displaystyle X\,}
2280:
2265:
2247:
2226:
1919:{\displaystyle Y\,}
1476:Hermitian conjugate
1259:*-regular semigroup
551:(by analogy with a
534:, as all bands are.
435:conjugate transpose
331:semigroup then the
3511:David James Foulis
3434:. You can help by
3381:
3352:
3292:
3246:
3175:
3145:
3100:
3046:
2988:
2930:
2891:
2823:
2791:
2766:
2738:
2710:
2682:
2661:
2633:
2600:
2565:
2541:
2486:
2447:
2427:
2392:
2365:
2319:
2284:
2266:
2251:
2227:
2212:
2179:
2151:
2099:
2079:
2056:
2020:
1944:
1916:
1891:
1860:
1855:
1727:
1707:
1669:
1649:
1618:symmetric alphabet
1591:
1570:
1434:
1408:
992:inverse semigroups
957:
814:partial isometries
106:full linear monoid
31:, particularly in
3996:978-1-84628-127-3
3968:978-0-521-16849-6
3936:978-3-11-015669-0
3911:978-3-642-01492-5
3870:978-0-8218-0627-2
3822:978-981-02-4042-4
3684:978-1-4612-1128-0
3646:978-3-211-82971-4
3621:978-1-4704-1493-1
3549:bounded operators
3530:quantum mechanics
3507:Foulis semigroups
3452:
3451:
3415:Baer *-semigroups
3405:involutive monoid
3393:Shamir congruence
2928:
2901:exists such that
2851:
2748:(and likewise on
2461:, the involution
2301:on the semigroup
2102:{\displaystyle Y}
2030:for some letters
1835:
1793:
1373:Michael P. Drazin
1251:
1250:
1205:inverse semigroup
1002:of *-semigroups:
988:previous examples
729:*. Equivalently,
695:partially ordered
589:regular semigroup
581:inverse semigroup
532:regular semigroup
454:converse relation
395:Underlying every
390:inverse semigroup
386:regular semigroup
375:inverse semigroup
201:Formal definition
180:converse relation
52:anti-automorphism
47:equipped with an
16:(Redirected from
4166:
4159:Semigroup theory
4055:World Scientific
4041:
4030:
4021:
4013:. 6 April 2013.
4007:
4001:
4000:
3982:
3973:
3972:
3954:
3941:
3940:
3922:
3916:
3915:
3893:
3887:
3884:
3875:
3874:
3856:
3845:
3844:
3836:
3827:
3826:
3808:
3799:
3796:
3787:
3784:
3778:
3775:
3769:
3766:
3760:
3757:
3751:
3748:
3742:
3739:
3728:
3725:
3719:
3716:
3710:
3707:
3698:
3695:
3689:
3688:
3670:
3664:
3659:H.S.M. Coxeter,
3657:
3651:
3650:
3632:
3626:
3625:
3607:
3564:coordinatization
3447:
3444:
3426:
3419:
3390:
3388:
3387:
3382:
3361:
3359:
3358:
3353:
3327:
3326:
3301:
3299:
3298:
3293:
3255:
3253:
3252:
3247:
3221:
3220:
3184:
3182:
3181:
3176:
3171:
3170:
3154:
3152:
3151:
3146:
3144:
3143:
3109:
3107:
3106:
3101:
3099:
3098:
3089:
3088:
3055:
3053:
3052:
3047:
3045:
3044:
3035:
3034:
2997:
2995:
2994:
2989:
2987:
2986:
2977:
2976:
2939:
2937:
2936:
2931:
2929:
2921:
2900:
2898:
2897:
2892:
2884:
2883:
2874:
2873:
2852:
2844:
2832:
2830:
2829:
2824:
2800:
2798:
2797:
2792:
2775:
2773:
2772:
2767:
2764:
2763:
2747:
2745:
2744:
2739:
2736:
2735:
2719:
2717:
2716:
2711:
2708:
2707:
2691:
2689:
2688:
2683:
2670:
2668:
2667:
2662:
2642:
2640:
2639:
2634:
2626:
2625:
2609:
2607:
2606:
2601:
2598:
2597:
2577:identity element
2574:
2572:
2571:
2566:
2550:
2548:
2547:
2542:
2528:
2527:
2515:
2514:
2495:
2493:
2492:
2487:
2484:
2483:
2456:
2454:
2453:
2448:
2436:
2434:
2433:
2428:
2426:
2425:
2401:
2399:
2398:
2393:
2390:
2389:
2384:
2374:
2372:
2371:
2366:
2364:
2363:
2354:
2353:
2328:
2326:
2325:
2320:
2317:
2316:
2293:
2291:
2290:
2285:
2279:
2274:
2264:
2259:
2246:
2241:
2225:
2220:
2208:
2207:
2188:
2186:
2185:
2180:
2177:
2176:
2160:
2158:
2157:
2152:
2150:
2149:
2137:
2136:
2124:
2123:
2118:
2108:
2106:
2105:
2100:
2088:
2086:
2085:
2080:
2065:
2063:
2062:
2057:
2046:
2045:
2029:
2027:
2026:
2021:
2019:
2018:
2006:
2005:
1993:
1992:
1983:
1982:
1953:
1951:
1950:
1945:
1942:
1941:
1925:
1923:
1922:
1917:
1900:
1898:
1897:
1892:
1889:
1888:
1869:
1867:
1866:
1861:
1859:
1858:
1852:
1851:
1836:
1833:
1820:
1819:
1794:
1791:
1764:
1763:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1691:
1678:
1676:
1675:
1670:
1659:be a bijection;
1658:
1656:
1655:
1650:
1648:
1647:
1600:
1598:
1597:
1592:
1579:
1577:
1576:
1571:
1569:
1568:
1445:
1443:
1441:
1440:
1435:
1433:
1419:
1417:
1415:
1414:
1409:
1407:
1370:
1351:
1339:
1325:
1287:Green's relation
1246:
1243:
1225:
1218:
966:
964:
963:
958:
950:
949:
931:
930:
763:ordered groupoid
737:if and only if
600:partial isometry
553:Hermitian matrix
476:rectangular band
446:binary relations
308:antidistributive
176:binary relations
129:
81:An example from
33:abstract algebra
21:
4174:
4173:
4169:
4168:
4167:
4165:
4164:
4163:
4144:
4143:
4116:Semigroup Forum
4110:Miyuki Yamada,
4105:Semigroup Forum
4050:
4045:
4044:
4031:
4024:
4008:
4004:
3997:
3984:
3983:
3976:
3969:
3956:
3955:
3944:
3937:
3924:
3923:
3919:
3912:
3895:
3894:
3890:
3885:
3878:
3871:
3858:
3857:
3848:
3838:
3837:
3830:
3823:
3810:
3809:
3802:
3797:
3790:
3785:
3781:
3776:
3772:
3767:
3763:
3758:
3754:
3749:
3745:
3740:
3731:
3726:
3722:
3717:
3713:
3708:
3701:
3696:
3692:
3685:
3672:
3671:
3667:
3658:
3654:
3647:
3634:
3633:
3629:
3622:
3609:
3608:
3604:
3599:
3580:Dagger category
3576:
3519:
3494:The projection
3448:
3442:
3439:
3432:needs expansion
3417:
3397:free half group
3364:
3363:
3318:
3304:
3303:
3266:
3265:
3258:Dyck congruence
3212:
3198:
3197:
3162:
3157:
3156:
3135:
3127:
3126:
3090:
3080:
3066:
3065:
3036:
3026:
3012:
3011:
2978:
2968:
2942:
2941:
2903:
2902:
2875:
2865:
2838:
2837:
2803:
2802:
2782:
2781:
2755:
2750:
2749:
2727:
2722:
2721:
2699:
2694:
2693:
2673:
2672:
2652:
2651:
2617:
2612:
2611:
2589:
2584:
2583:
2556:
2555:
2519:
2506:
2501:
2500:
2475:
2470:
2469:
2463:of every letter
2439:
2438:
2417:
2412:
2411:
2382:
2377:
2376:
2355:
2345:
2331:
2330:
2308:
2303:
2302:
2297:This map is an
2199:
2194:
2193:
2168:
2163:
2162:
2141:
2128:
2116:
2111:
2110:
2091:
2090:
2071:
2070:
2037:
2032:
2031:
2010:
1997:
1984:
1974:
1963:
1962:
1933:
1928:
1927:
1907:
1906:
1880:
1875:
1874:
1854:
1853:
1843:
1830:
1808:
1805:
1804:
1788:
1769:
1755:
1750:
1749:
1719:
1718:
1685:
1684:
1683:to a bijection
1661:
1660:
1639:
1622:
1621:
1614:with involution
1582:
1581:
1560:
1543:
1542:
1504:
1489:
1461:
1424:
1423:
1421:
1398:
1397:
1395:
1353:
1341:
1327:
1312:
1247:
1241:
1238:
1231:needs expansion
1214:
1134:
1054:D. B. McAlister
996:Boris M. Schein
984:
976:
941:
922:
911:
910:
904:
884:is the meet of
876:and kernel the
845:if and only if
824:
799:
767:partial product
682:if and only if
583:if and only if
541:
411:
403:is the algebra
320:
219:unary operation
203:
158:), with string
150:generated by a
146:theory, is the
144:formal language
140:diagonal matrix
117:
23:
22:
15:
12:
11:
5:
4172:
4170:
4162:
4161:
4156:
4146:
4145:
4140:
4139:
4126:
4119:
4108:
4101:
4096:Drazin, M.P.,
4094:
4075:
4064:
4049:
4046:
4043:
4042:
4022:
4002:
3995:
3974:
3967:
3942:
3935:
3917:
3910:
3888:
3876:
3869:
3846:
3828:
3821:
3800:
3788:
3779:
3770:
3761:
3752:
3750:Lawson, p. 118
3743:
3741:Lawson, p. 117
3729:
3727:Lawson, p. 116
3720:
3711:
3699:
3690:
3683:
3665:
3652:
3645:
3627:
3620:
3601:
3600:
3598:
3595:
3594:
3593:
3588:
3583:
3575:
3572:
3518:
3515:
3492:
3491:
3450:
3449:
3429:
3427:
3416:
3413:
3380:
3377:
3374:
3371:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3325:
3321:
3317:
3314:
3311:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3219:
3215:
3211:
3208:
3205:
3174:
3169:
3165:
3142:
3138:
3134:
3097:
3093:
3087:
3083:
3079:
3076:
3073:
3043:
3039:
3033:
3029:
3025:
3022:
3019:
2985:
2981:
2975:
2971:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2927:
2924:
2919:
2916:
2913:
2910:
2890:
2887:
2882:
2878:
2872:
2868:
2864:
2861:
2858:
2855:
2850:
2847:
2822:
2819:
2816:
2813:
2810:
2789:
2762:
2758:
2734:
2730:
2706:
2702:
2680:
2659:
2643:, we obtain a
2632:
2629:
2624:
2620:
2596:
2592:
2575:(which is the
2563:
2540:
2537:
2534:
2531:
2526:
2522:
2518:
2513:
2509:
2482:
2478:
2446:
2424:
2420:
2388:
2362:
2358:
2352:
2348:
2344:
2341:
2338:
2315:
2311:
2295:
2294:
2283:
2278:
2273:
2269:
2263:
2258:
2254:
2250:
2245:
2240:
2237:
2234:
2230:
2224:
2219:
2215:
2211:
2206:
2202:
2175:
2171:
2148:
2144:
2140:
2135:
2131:
2127:
2122:
2098:
2078:
2069:The bijection
2067:
2066:
2055:
2052:
2049:
2044:
2040:
2017:
2013:
2009:
2004:
2000:
1996:
1991:
1987:
1981:
1977:
1973:
1970:
1940:
1936:
1914:
1903:free semigroup
1887:
1883:
1873:Now construct
1871:
1870:
1857:
1850:
1846:
1842:
1839:
1831:
1829:
1826:
1823:
1818:
1815:
1811:
1807:
1806:
1803:
1800:
1797:
1789:
1787:
1784:
1781:
1778:
1775:
1774:
1772:
1767:
1762:
1758:
1726:
1706:
1703:
1700:
1697:
1694:
1668:
1646:
1642:
1638:
1635:
1632:
1629:
1603:disjoint union
1589:
1567:
1563:
1559:
1556:
1553:
1550:
1516:free semigroup
1503:
1500:
1485:
1457:
1451:multiplicative
1432:
1406:
1379:, the element
1249:
1248:
1228:
1226:
1213:
1210:
1200:
1199:
1184:
1173:
1133:
1130:
1045:
1044:
1017:
983:
980:
975:
972:
956:
953:
948:
944:
940:
937:
934:
929:
925:
921:
918:
900:
820:
798:
795:
602:is an element
540:
537:
536:
535:
472:
461:
438:
437:as involution.
407:
393:
367:
336:
319:
316:
312:H.S.M. Coxeter
292:The semigroup
290:
289:
262:
202:
199:
148:free semigroup
100:of order
87:multiplicative
83:linear algebra
76:
75:
72:
69:
60:unary operator
24:
14:
13:
10:
9:
6:
4:
3:
2:
4171:
4160:
4157:
4155:
4152:
4151:
4149:
4142:
4138:
4136:
4132:
4127:
4124:
4120:
4117:
4113:
4109:
4106:
4102:
4099:
4095:
4092:
4088:
4084:
4080:
4076:
4073:
4069:
4065:
4063:
4062:981-02-3316-7
4059:
4056:
4052:
4051:
4047:
4039:
4035:
4029:
4027:
4023:
4020:
4016:
4012:
4006:
4003:
3998:
3992:
3988:
3981:
3979:
3975:
3970:
3964:
3960:
3953:
3951:
3949:
3947:
3943:
3938:
3932:
3928:
3921:
3918:
3913:
3907:
3903:
3899:
3892:
3889:
3886:Lawson p. 172
3883:
3881:
3877:
3872:
3866:
3862:
3855:
3853:
3851:
3847:
3842:
3835:
3833:
3829:
3824:
3818:
3814:
3807:
3805:
3801:
3795:
3793:
3789:
3783:
3780:
3774:
3771:
3765:
3762:
3756:
3753:
3747:
3744:
3738:
3736:
3734:
3730:
3724:
3721:
3715:
3712:
3706:
3704:
3700:
3697:Munn, Lemma 1
3694:
3691:
3686:
3680:
3676:
3669:
3666:
3662:
3656:
3653:
3648:
3642:
3638:
3631:
3628:
3623:
3617:
3613:
3606:
3603:
3596:
3592:
3589:
3587:
3584:
3581:
3578:
3577:
3573:
3571:
3569:
3565:
3560:
3558:
3554:
3550:
3546:
3545:Hilbert space
3542:
3537:
3535:
3531:
3526:
3524:
3516:
3514:
3512:
3508:
3503:
3501:
3497:
3489:
3485:
3481:
3477:
3473:
3472:
3471:
3469:
3465:
3461:
3457:
3446:
3437:
3433:
3430:This section
3428:
3425:
3421:
3420:
3414:
3412:
3410:
3407:generated by
3406:
3402:
3398:
3394:
3378:
3375:
3346:
3343:
3340:
3337:
3331:
3328:
3323:
3319:
3315:
3289:
3286:
3277:
3263:
3262:Dyck language
3259:
3240:
3237:
3234:
3231:
3225:
3222:
3217:
3213:
3209:
3196:
3192:
3188:
3172:
3167:
3163:
3140:
3136:
3132:
3124:
3120:
3116:
3111:
3095:
3085:
3081:
3077:
3074:
3063:
3059:
3041:
3031:
3027:
3023:
3020:
3009:
3008:diagram order
3005:
3001:
3000:inclusion map
2983:
2973:
2969:
2965:
2962:
2953:
2950:
2947:
2917:
2914:
2911:
2888:
2880:
2870:
2866:
2862:
2859:
2853:
2836:
2820:
2814:
2811:
2787:
2779:
2760:
2756:
2732:
2728:
2704:
2700:
2678:
2657:
2648:
2646:
2630:
2627:
2622:
2618:
2594:
2590:
2582:
2578:
2561:
2554:
2535:
2529:
2524:
2520:
2516:
2511:
2507:
2499:
2480:
2476:
2466:
2464:
2460:
2444:
2422:
2418:
2409:
2405:
2386:
2375:with the map
2360:
2350:
2346:
2342:
2339:
2313:
2309:
2300:
2281:
2276:
2271:
2267:
2261:
2256:
2252:
2248:
2243:
2238:
2235:
2232:
2228:
2222:
2217:
2213:
2209:
2204:
2200:
2192:
2191:
2190:
2173:
2169:
2146:
2142:
2133:
2129:
2125:
2120:
2096:
2076:
2053:
2050:
2047:
2042:
2038:
2015:
2011:
2007:
2002:
1998:
1994:
1989:
1985:
1979:
1975:
1971:
1968:
1961:
1960:
1959:
1957:
1956:concatenation
1938:
1934:
1912:
1904:
1885:
1881:
1848:
1844:
1840:
1837:
1824:
1816:
1813:
1809:
1801:
1798:
1795:
1782:
1776:
1770:
1765:
1760:
1756:
1748:
1747:
1746:
1744:
1740:
1724:
1704:
1698:
1695:
1692:
1682:
1679:is naturally
1666:
1644:
1640:
1633:
1630:
1627:
1619:
1615:
1613:
1608:
1604:
1587:
1565:
1561:
1557:
1554:
1551:
1548:
1540:
1536:
1535:disjoint sets
1532:
1528:
1523:
1521:
1517:
1513:
1509:
1501:
1499:
1497:
1493:
1488:
1484:
1480:
1477:
1473:
1469:
1465:
1460:
1456:
1452:
1447:
1392:
1390:
1386:
1382:
1378:
1374:
1369:
1365:
1361:
1357:
1349:
1345:
1338:
1334:
1330:
1323:
1319:
1315:
1310:
1306:
1302:
1298:
1296:
1291:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1245:
1236:
1232:
1229:This section
1227:
1224:
1220:
1219:
1216:
1211:
1209:
1206:
1197:
1193:
1189:
1185:
1182:
1178:
1174:
1171:
1167:
1163:
1159:
1155:
1151:
1150:
1149:
1147:
1143:
1139:
1131:
1129:
1127:
1123:
1119:
1115:
1111:
1106:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1071:
1067:
1063:
1058:
1055:
1051:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1016:
1012:
1008:
1005:
1004:
1003:
1001:
997:
993:
989:
981:
979:
973:
971:
968:
954:
946:
942:
938:
935:
932:
927:
923:
916:
908:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
851:
848:
844:
840:
836:
832:
828:
823:
819:
815:
810:
808:
804:
796:
794:
792:
788:
784:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
691:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
625:
621:
617:
613:
609:
605:
601:
597:
592:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
538:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
470:
466:
462:
459:
455:
451:
447:
443:
439:
436:
432:
431:
426:
423:
419:
415:
410:
406:
402:
398:
394:
391:
387:
383:
380:
376:
372:
368:
365:
364:abelian group
361:
357:
353:
349:
345:
341:
337:
334:
330:
326:
322:
321:
317:
315:
313:
309:
304:
302:
297:
295:
287:
283:
279:
275:
271:
267:
263:
260:
256:
252:
248:
244:
243:
242:
240:
236:
232:
228:
224:
220:
216:
212:
208:
200:
198:
196:
192:
191:Viktor Wagner
187:
185:
181:
177:
173:
169:
165:
161:
160:concatenation
157:
153:
149:
145:
141:
137:
133:
128:
125:
121:
115:
111:
107:
103:
99:
95:
91:
88:
84:
79:
73:
70:
68:
65:
64:
63:
61:
57:
53:
50:
46:
42:
38:
34:
30:
19:
4141:
4128:
4111:
4097:
4082:
4078:
4067:
4010:
4005:
3986:
3958:
3926:
3920:
3901:
3898:Arto Salomaa
3891:
3860:
3840:
3812:
3798:Lawson p. 51
3782:
3773:
3768:Lawson p.120
3764:
3755:
3746:
3723:
3714:
3693:
3674:
3668:
3660:
3655:
3636:
3630:
3611:
3605:
3561:
3552:
3540:
3538:
3534:Baer *-rings
3527:
3520:
3506:
3504:
3499:
3495:
3493:
3487:
3483:
3479:
3475:
3467:
3463:
3459:
3453:
3440:
3436:adding to it
3431:
3408:
3404:
3396:
3392:
3257:
3119:reduced word
3112:
3006:is taken in
2649:
2644:
2467:
2462:
2407:
2403:
2296:
2068:
1872:
1617:
1610:
1606:
1531:equinumerous
1524:
1512:free objects
1505:
1495:
1491:
1486:
1482:
1478:
1471:
1467:
1463:
1458:
1454:
1448:
1393:
1384:
1380:
1376:
1367:
1363:
1359:
1355:
1347:
1343:
1336:
1332:
1328:
1321:
1317:
1313:
1311:′ such that
1308:
1304:
1300:
1294:
1289:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1253:A semigroup
1252:
1239:
1235:adding to it
1230:
1215:
1201:
1195:
1191:
1187:
1180:
1176:
1169:
1165:
1161:
1160:) such that
1157:
1153:
1145:
1141:
1137:
1135:
1126:I-semigroups
1121:
1117:
1113:
1109:
1107:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1065:
1061:
1059:
1050:free objects
1046:
1040:
1036:
1032:
1028:
1024:
1020:
1014:
1010:
1006:
985:
977:
969:
906:
901:
897:
889:
885:
881:
873:
869:
865:
861:
857:
853:
849:
842:
838:
834:
830:
826:
821:
817:
811:
803:difunctional
800:
790:
786:
782:
778:
774:
770:
758:
754:
750:
746:
742:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
692:
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
603:
599:
593:
584:
576:
572:
568:
564:
560:
556:
548:
544:
542:
527:
523:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
479:
449:
441:
428:
421:
417:
413:
408:
404:
370:
359:
355:
351:
347:
339:
333:identity map
324:
305:
301:U-semigroups
298:
293:
291:
285:
281:
277:
273:
269:
265:
258:
254:
250:
246:
238:
234:
230:
226:
222:
214:
206:
204:
188:
168:linear order
152:nonempty set
135:
126:
123:
119:
104:(called the
101:
80:
77:
40:
36:
26:
4077:W.D. Munn,
3896:Ion Petre;
3456:right ideal
3058:isomorphism
2498:free monoid
2496:we use the
1172:are in F(S)
894:semilattice
872:with image
543:An element
469:free monoid
433:, with the
379:idempotents
354:defined by
329:commutative
41:*-semigroup
29:mathematics
4148:Categories
4131:PlanetMath
4091:0521576695
4048:References
3470:such that
3443:April 2015
3401:Eli Shamir
3195:congruence
3115:free group
2801:and a map
2553:empty word
2299:involution
1745:notation:
1527:generators
1520:free group
1453:semigroup
1242:April 2015
1000:subvariety
628:projection
606:such that
397:C*-algebra
172:set theory
67:Uniqueness
49:involutive
3586:*-algebra
3523:example 5
3379:ε
3344:∈
3332:ε
3324:†
3290:ε
3238:∈
3226:ε
3218:†
3187:confluent
3168:†
3141:†
3123:rewriting
3096:∗
3086:†
3078:⊔
3032:†
3024:⊔
2974:†
2966:⊔
2957:→
2948:ι
2926:¯
2923:Φ
2918:∘
2915:ι
2909:Φ
2886:→
2871:†
2863:⊔
2849:¯
2846:Φ
2833:, then a
2818:→
2809:Φ
2761:∗
2705:†
2658:θ
2631:ε
2623:†
2619:ε
2595:∗
2562:ε
2536:ε
2530:∪
2512:∗
2459:Example 6
2445:θ
2423:†
2387:†
2351:†
2343:⊔
2277:†
2262:†
2249:⋯
2244:†
2236:−
2223:†
2205:†
2139:→
2121:†
2077:†
2048:∈
2008:∈
1995:⋯
1849:†
1841:∈
1814:−
1810:θ
1799:∈
1777:θ
1761:†
1743:piecewise
1725:θ
1702:→
1693:†
1667:θ
1645:†
1637:→
1628:θ
1588:⊔
1566:†
1558:⊔
1194:in F(S),
1132:P-systems
1070:Example 7
947:∗
936:∧
928:∗
769:given by
765:with the
549:hermitian
382:invariant
276:we have (
211:semigroup
195:semiheaps
114:transpose
45:semigroup
3574:See also
3509:, after
3486:= 0 } =
3191:quotient
2940:, where
1834:if
1792:if
1741:, or in
1681:extended
1612:alphabet
1508:category
1281:, where
1186:For any
1175:For any
1152:For any
1138:P-system
797:Examples
761:) is an
425:matrices
401:instance
318:Examples
264:For all
245:For all
164:reverses
156:alphabet
98:matrices
3663:, p. 33
3557:adjoint
2998:is the
2579:of the
1901:as the
1739:inverse
1474:to its
1449:In the
1444:
1422:
1418:
1396:
1285:is the
773:⋅
465:strings
108:). The
96:square
85:is the
4089:
4060:
3993:
3965:
3933:
3908:
3867:
3819:
3681:
3643:
3618:
3121:and a
2581:monoid
1954:being
1620:. Let
1346:′)* =
1297:-class
1164:° and
579:is an
522:)* = (
388:is an
373:is an
257:*)* =
90:monoid
3597:Notes
3543:is a
2671:from
1616:or a
1362:)* =
1273:* is
1112:** =
1097:) = (
1031:) = (
626:). A
587:is a
567:* or
506:) = (
427:over
416:) of
344:group
342:is a
327:is a
280:)* =
221:* on
217:is a
209:be a
56:group
43:is a
39:or a
4087:ISBN
4058:ISBN
3991:ISBN
3963:ISBN
3931:ISBN
3906:ISBN
3865:ISBN
3817:ISBN
3679:ISBN
3641:ISBN
3616:ISBN
3002:and
1525:The
1420:and
1320:′ =
1181:a°ba
888:and
852:⊆ im
833:and
812:The
745:and
721:* =
717:and
670:and
662:and
646:* =
642:and
598:. A
559:* =
420:-by-
358:* =
205:Let
166:the
154:(an
122:) =
94:real
35:, a
4034:doi
4015:doi
3566:of
3551:on
3539:If
3438:.
3155:or
2692:to
2406:on
2089:on
1905:on
1537:in
1303:in
1265:in
1237:.
1105:).
1023:*)(
880:of
793:*.
781:if
751:ett
697:by
448:on
440:If
369:If
338:If
323:If
272:in
253:, (
249:in
233:,
110:map
92:of
27:In
4150::
4114:,
4025:^
3977:^
3945:^
3879:^
3849:^
3831:^
3803:^
3791:^
3732:^
3702:^
3570:.
3559:.
3536:.
3502:.
3488:eS
3484:xy
3482:|
3478:∈
3474:{
3462:∈
2647:.
1958::
1541::
1533:)
1498:.
1371:.
1352:,
1348:xx
1344:xx
1340:,
1335:=
1329:xx
1326:,
1318:xx
1269:,
1196:ab
1190:,
1162:aa
1128:.
1118:xx
1116:=
1101:,
1093:,
1089:)(
1085:,
1077:,
1043:*)
1041:xx
1039:)(
1021:xx
1011:xx
1009:=
990:,
967:.
864:=
860:∩
847:im
841:≤
809:.
791:tt
789:=
779:st
777:=
749:=
743:et
741:=
733:≤
727:tt
723:ss
719:ss
711:ss
709:=
701:≤
690:.
688:fe
686:=
680:ef
678:=
664:ss
654:,
638:=
636:ee
614:=
608:ss
565:xx
526:,
518:,
510:,
502:,
498:)(
494:,
486:×
474:A
350:→
303:.
288:*.
278:xy
268:,
237:↦
229:→
197:.
186:.
136:AA
120:AB
4137:.
4040:.
4036::
4017::
3999:.
3971:.
3939:.
3914:.
3873:.
3825:.
3687:.
3649:.
3624:.
3553:H
3541:H
3500:x
3496:e
3490:.
3480:S
3476:y
3468:e
3464:S
3460:x
3445:)
3441:(
3409:X
3376:=
3373:)
3370:(
3350:}
3347:X
3341:x
3338::
3335:)
3329:,
3320:x
3316:x
3313:(
3310:{
3287:=
3284:(
3281:)
3278:=
3275:)
3272:(
3244:}
3241:Y
3235:y
3232::
3229:)
3223:,
3214:y
3210:y
3207:(
3204:{
3173:x
3164:x
3137:x
3133:x
3092:)
3082:X
3075:X
3072:(
3042:+
3038:)
3028:X
3021:X
3018:(
2984:+
2980:)
2970:X
2963:X
2960:(
2954:X
2951::
2912:=
2889:S
2881:+
2877:)
2867:X
2860:X
2857:(
2854::
2821:S
2815:X
2812::
2788:S
2757:Y
2733:+
2729:Y
2701:X
2679:X
2628:=
2591:Y
2539:}
2533:{
2525:+
2521:Y
2517:=
2508:Y
2481:+
2477:Y
2419:X
2408:X
2361:+
2357:)
2347:X
2340:X
2337:(
2314:+
2310:Y
2282:.
2272:1
2268:w
2257:2
2253:w
2239:1
2233:k
2229:w
2218:k
2214:w
2210:=
2201:w
2174:+
2170:Y
2147:+
2143:Y
2134:+
2130:Y
2126::
2097:Y
2054:.
2051:Y
2043:i
2039:w
2016:+
2012:Y
2003:k
1999:w
1990:2
1986:w
1980:1
1976:w
1972:=
1969:w
1939:+
1935:Y
1913:Y
1886:+
1882:Y
1845:X
1838:y
1828:)
1825:y
1822:(
1817:1
1802:X
1796:y
1786:)
1783:y
1780:(
1771:{
1766:=
1757:y
1705:Y
1699:Y
1696::
1641:X
1634:X
1631::
1607:Y
1562:X
1555:X
1552:=
1549:Y
1496:A
1492:C
1490:(
1487:n
1483:M
1479:A
1472:A
1468:n
1464:C
1462:(
1459:n
1455:M
1431:C
1405:R
1385:x
1381:x
1377:x
1368:x
1366:′
1364:x
1360:x
1358:′
1356:x
1354:(
1350:′
1342:(
1337:x
1333:x
1331:′
1324:′
1322:x
1316:′
1314:x
1309:x
1305:S
1301:x
1295:L
1290:H
1283:H
1279:x
1275:H
1271:x
1267:S
1263:x
1255:S
1244:)
1240:(
1192:b
1188:a
1177:a
1170:a
1168:°
1166:a
1158:a
1154:a
1146:a
1142:a
1122:x
1120:*
1114:x
1110:x
1103:b
1099:a
1095:b
1091:b
1087:a
1083:a
1079:x
1075:x
1073:(
1066:x
1062:x
1037:x
1035:*
1033:x
1029:x
1027:*
1025:x
1019:(
1015:x
1013:*
1007:x
955:B
952:)
943:B
939:B
933:A
924:A
920:(
917:A
907:C
905:(
902:n
898:M
890:F
886:E
882:V
874:V
870:J
866:V
862:F
858:E
854:F
850:E
843:F
839:E
835:F
831:E
827:C
825:(
822:n
818:M
787:s
785:*
783:s
775:t
771:s
759:S
755:e
747:e
739:s
735:t
731:s
725:*
715:t
713:*
707:s
703:t
699:s
684:e
676:e
672:f
668:e
660:s
658:*
656:s
652:s
648:e
644:e
640:e
632:e
624:S
620:S
616:s
612:s
610:*
604:s
585:S
577:S
573:x
571:*
569:x
561:x
557:x
545:x
528:a
524:b
520:b
516:a
512:d
508:a
504:d
500:c
496:b
492:a
488:A
484:A
480:A
450:X
442:X
430:C
422:n
418:n
414:C
412:(
409:n
405:M
371:S
360:x
356:x
352:S
348:S
340:S
325:S
294:S
286:x
284:*
282:y
274:S
270:y
266:x
261:.
259:x
255:x
251:S
247:x
239:x
235:x
231:S
227:S
223:S
215:S
207:S
127:A
124:B
118:(
102:n
20:)
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