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Научни трудове на Съюза на учените в България-Пловдив, серия Б. Естествени и хуманитарни науки, т. XV, 2013 г. Научна сесия „Техника и технологии, естествени и хуманитарни науки", 25-26 X 2012 Scientific researches of the Union of Scientists in Bulgaria-Plovdiv, series B. Natural Sciences and the Humanities, Vol. XV, ISSN 1311-9192, Technics, Technologies, Natural Sciences and Humanities Session, 25-26 oktober 2012.
CONNECTIONS BETWEEN HOMOMORPHISMS AND CONGRUENCES IN REGULAR SEMIGRUOPS AND INVERSE
SEMIGROUPS
Sabri Sadikua a-FGJT; University of Prishtina, Mitrovica, Republic of Kosovo
ABSTRACT
In this paper we initially define the semigroup, regular semigroup, and inverse semigroup. Furthermore, we define homomorphism and congruence in a semigroup. These notions are already known in the theory of semigroups. The new finding in this paper is the proof that if S is a regular
semigroup and T is a semigroup, and if a homomorfism Ф exists from S in T , then Sф is a regular semigroup. Additionally, if S is an inverse semigroup, and T a semigroup, and if a homomorfism Ф
exists from S in T , then Sф is a inverse semigroup. All these are proved in Lemma 1 and Theorem 1. Furthermore, with Theorem 2and 3 we establish the connection between a congruence p its
Key words: Regular semigroup; inverse semigroup, homomorphism, congruence.
Introduction: Initially we defined grupoid and semigroup, notions which are already known in the theory of
semigroups. But we first define the binary operation. Let S be an arbitrary set. A mapping from S x S in S is
called binary operation in set S and is noted by " •". For binary operation " •" in set S we say it is associative if for
a, b, c e S is
a • (b • c) = (a • b) • c .
Now we can define the grupoid and semigroup.
Definition 1. A set S together with a binary operation is called a grupoid and is noted by
(S, •) . A grupoid S
satisfying the associative law:
a • (b • c) = (a • b) • c, for a, b, c e S , is a semigroup and is also noted by ( S, •) .
The regular element, regular semigroup and inverse element are defined in [2] by J.M.Howie.
Definition 2. Element a e S is regular element, if element x e S exists, which satisfies equation a = axa .
A semigroup S is a regular semigroup if all its elements are regular.
Element a- e S is inverse element of a e S if a = aa~la and a-1 = a-1 aa~1. Now after defining regular element and inverse element we can define inverse semigroup.
Definition 3. Semigroup S is inverse semigroup if for every element a e S exists one and only one inverse element
a-1 e S.
Definition of homomorphism in semigroup is taken from [1] M.Petrich.(p.17,def.I.4.1.).
Definition 4. Let S and Tbe two semigroups. Mapping (j) .S ^ T is homomorphism if ^ satisfies relation
(ab)( = (a()(b(), per a, b e S .
Definition of congruence in semigroup S is taken from [3] S.Sadiku (p.25. def. 2), Definition 5. Relation p on the semigroup S is called left compatible if
(Va,b,c e S);(a,b) ep^ (ca,cb) e p
and p is right compatible if
(V a, b, c e S); (a, b) ep^ (ac, bc) e p
A left (right) compatible equivalence relation on the semigroup S is called left (right) congruence.
A equivalence relation on the semigroup S is congruence if is left and right congruence.
If p is a congruence on the semigroup S, then quotient set with binary operation (ap)(bp) = (ab)p; Va, b e S
is semigroup. Here we prove only associative law :
for a, b, c e S and ap, bp, cpe Sp we have (ap)[(bp)(cp)] = (ap)[(bc)p] = [a(bc)]p = [(ab)c]p = [(ab)p](cp) = [(ap)(bp)](cp)
We now take two results, which characterize connection between homomorphism and congruence in regular semigroups.
Lema 1. Let S be an regular semigroup and T semigroup. If mapping ( : S ^ T
is homomorphism from S in T, then S( is a regular semigroup. Proof. Let S be an regular semigroup, T semigroup and ( homomorphism
( : S ^ T .
Let a e S and a' = a( one element from S(. Since S is a regular semigroup it implies that element x e S exists for which is a = axa and x = xax. Let x' = x(, x' e S( . Following relations are true in S( :
a' x' a' = (a()( x()(a() = (axa)( = a( = a' x' a' x' = (x()(a()( x() = (xax)( = x( = x'.
Since for arbitrary element a' from S( inverse element x' e S( exists, we conclude that S( is a regular semigroup.
Now we take Theorem 11.2.1. from [1] which is needed to prove the following theorems. For
the sake of correctness we write this theorem.
Theorem II.2.1.[1]. The following conditions on semigroup S are equivalent : (i) S is inverse semigoup
(ii) S is regular semigoup and idempotents commutes.
Theorem 1. Let S be an inverse semigroup and T semigroup. If mapping
<:S ^ T
is homomorphism from S in T, then S< is inverse semigroup.
Proof. From Lema 1. Implies that S< is a regular semigroup. From Llallement Lema (Lema II.4.7.) [2] idempotent e and f from S exist, for which is e< = g and f < = h , where g and h are idempotent in S<. Based in Theorem II.1.2. [1], since S< is a regular semigroup, it's enough to show that idempotents g, h e S< commute . Indeed
gh = (e<)(f<) = (ef )< = (fe)< = (f<)(e<) = hg.
from this we conclude that S< is inverse semigroup.
To prove one characteristic of inverse semigroup as formulated in the following lema.
Lema 2. F or every element 5 e S , element se S< is inverse element for s< e S<, where < is homomorphism < : S ^ S< .
Proof. Let S be an inverse semigroup and < homomorphism from S on S<. For s e S is
(s<)( s - <)(s<) = (ss -1 s)< = s<
and
(s ~V)(s<)(s - <) = (s-1 ss -1 )< = s
Since S< is inverse semigroup, then inverse element is unique, exactly
(s<)-1 = (s-<).
Finally, we conclude that inverse element for element s( e S( is element s ( e S( .
Theorem 2. Let p be a congruence in regular semigroup S, then ( S/ , •) is regular semigroup and mapping \/ p '
ф: S - S/p
determined by
аф = ap, a e S is homomorphism.
Proof. First we show that ф is homomorphism. Let a, b e S , then (аЬ)ф = (ab)p = (ap)(bp) = (аф)(Ьф) .
From (aЬ)ф = (aф)(bф) ; a, b e S it is implied that ф is homomorphism. Finally from Lema 1. we conclude that
is a regular semigroup.
( S/p-) '
Theorem 3. Let p be a congruence in inverse semigroup S, then ( S/ , •) is inverse semigroup and mapping \/ p >
ф : S — S/ p
determined by
aф = ap, a e S is homomorphism.
Proof. From Theorem 2.it is implied that ф is homomorphism and ( S/ , •) is regular
semigroup. Finally from \/ p >
Theorem 1. we conclude that ( , •) is inverse semigoup.
Conclusion: We conclude that for every regular (inverse) semigroup S, Sф is regular
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(inverse) semigroup, where < is homomorphism. From the last two theorems we conclude that for
every regular (inverse) semigroups S, quotient semigroup ( , •) is regular(inverse) semigroup, where p is congruence in S and < homomorphism.
[1] M. Petrich, Inverse Semigroups, ISBN 0-471-875445-7, 1984 BY John Wiley&Sons.
[2] J.M.Howie, An introduction to semigroup theory, ISBN 75-46333, 1976 Academic press inc.(London) ltd
[3] S. Sadiku, Kongruencat ndarese te majta dhe te djathta te idempotenteve ne gjysmegrupet inversive, BSHN(UT), 2008,Nr.5, (f.25-35)
[4] S. Sadiku, Kongruencat ne gjysmegrupet inversive dhe r - gjysmegrupet inversive, Doctoral Dissertation, University of Tirana, 2010.
Рецензент: проф. д-р Иван Димовски
