Colloquium c
° 2012 AMSS CAS & SUZHOU UNIV
Gr¨
obner-Shirshov Bases of the Generalized
Bruck-Reilly ∗-Extension
Canan Kocapinar
Balikesir University, Department of Mathematics
Faculty of Art and Science, Cagis Campus, 10145, Balikesir, Turkey E-mail: canankocapinar@gmail.com
Eylem G¨uzel Karpuz
Karamanoglu Mehmetbey University, Department of Mathematics Faculty of Science, Yunus Emre Campus, 70100, Karaman, Turkey
E-mail: eguzel@balikesir.edu.tr
Firat Ate¸s†
Balikesir University, Department of Mathematics
Faculty of Art and Science, Cagis Campus, 10145, Balikesir, Turkey E-mail: firat@balikesir.edu.tr
A. Sinan C¸ evik
Sel¸cuk University, Department of Mathematics
Faculty of Science, Alaaddin Keykubat Campus, 42075, Konya, Turkey E-mail: sinan.cevik@selcuk.edu.tr
Received 6 December 2009 Communicated by L.A. Bokut
Abstract. In this paper we first define a presentation for the generalized Bruck-Reilly ∗-extension of a monoid and then we work on its Gr¨obner-Shirshov bases.
2000 Mathematics Subject Classification:20F05, 20M05
Keywords: Bruck-Reilly extension, Gr¨obner-Shirshov bases, monoid presentation 1 Introduction and Preliminaries
In combinatorial group and semigroup theory, for a finitely generated semigroup (monoid), a fundamental problem is to find its presentation with respect to some (irreducible) system of generators and relations and then investigate some algebraic and geometric properties of this semigroup (monoid). In this sense, in [15], the authors obtained a presentation for the Bruck-Reilly extension which was studied previously by Bruck [9], Munn [19] and Reilly [20]. In different manners, this exten-sion is considered as a fundamental construction in the theory of semigroups. Many classes of regular semigroups are characterized by Bruck-Reilly extensions; for in-stance, any bisimple regular w-semigroup is isomorphic to a Reilly extension of a
†Corresponding author.
group (see [20]) and any simple regular w-semigroup is isomorphic to a Bruck-Reilly extension of a finite chain of groups (see [17, 18]). In [1], the author obtained a new monoid, namely the generalized Bruck-Reilly ∗-extension, and then established the structure of the ∗-bisimple type A w-semigroup in which D∗= eD was obtained (see the end of this section for the definition of the equivalence relation on D). Using this, the w2-chain of idempotents was defined and then the structure theorem of the ∗-bisimple type A w2-semigroup as the generalized Bruck-Reilly ∗-extension was studied (see [21]). Moreover, in [16], the authors defined Gr¨obner-Shirshov bases for some monoid constructions. By considering these studies, our aim in this paper is to find a presentation of the generalized Bruck-Reilly ∗-extension and then obtain Gr¨obner-Shirshov bases of this special monoid.
The theory of Gr¨obner and Gr¨obner-Shirshov bases was invented independently by Shirshov [22] for non-commutative and non-associative algebras, and by Hironaka [14] and Buchberger [10] for commutative algebras. The technique of Gr¨obner-Shirshov bases is proved to be very useful in the study of presentations of associative algebras, Lie algebras, semigroups, groups and Ω-algebras by considering generators and relations (see, for example, the book [8] written by Bokut and Kukin, and survey papers [4] and [7]).
Let X be a set, X∗ the set of X-words (monomials), and < a monomial well ordering of X∗ (i.e., < is a well ordering that respects left and right multiplications by words). Also, let k be a field and khXi the free algebra over X and k. For f ∈ khXi, let f be the maximal (leading) monomial of f . Then f g = f g for any f, g. A polynomial f is called monic if the coefficient at f in f is equal to 1. Thus, for some monic polynomials f, g and a, b ∈ X∗, the Gr¨obner-Shirshov basis can be formulated as follows:
(I) Let w be a word such that w = f b = ag with deg(f ) + deg(g) > deg(w). Then the polynomial (f, g)wis called the intersection composition of f and g with respect to w if (f, g)w= f b − ag.
(II) Let w = f = agb. Then the polynomial (f, g)w = f − agb is called the
inclusion compositionof f and g with respect to w. In this case, the transformation f 7→ (f, g)w= f − agb is called the elimination of the leading word (ELW) of g in f . The first and second compositions given in (I) and (II) are denoted by f ∧ g and f ∨ g, respectively. Moreover, the word w is called the ambiguity of the composition (f, g)w. In addition, a composition (f, g)w is called trivial modulo (S, w), written as (f, g)w ≡ 0 mod (S, w), if (f, g)w =Pαiaisibi and aisibi < w, where si ∈ S, ai, bi ∈ X∗ and αi ∈ k. In particular, if (f, g)w is zero by ELW’s of polynomials from S, then (f, g)w≡ 0 mod (S, w).
Definition 1.1. A monic subset S of khXi is called a Gr¨obner-Shirshov basis (set) if any composition of polynomials from S is trivial modulo S and the ambiguity. In this case, S is also called a Gr¨obner-Shirshov basis of the ideal Id(S) generated by S and of the algebra khX; Si = khXi/Id(S) generated by X with defining relations S. The following lemma is due to Shirshov [22] (see also [2] and [3]). It is an analog of the main theorem of Buchberger [10, 11].
Lemma 1.2.(Composition-Diamond Lemma) Let S ⊂ khXi be a monic set and let < be a monomial well ordering of X∗. Then the following conditions are equivalent:
(1) S is a Gr¨obner-Shirshov basis relative to <.
(2) If f ∈ Id(S), then f = asb for some s ∈ S and a, b ∈ X∗.
(3) Irr(S) = {u : u 6= asb, s ∈ S, a, b ∈ X∗} is a k-base of the algebra khX; Si. If S is not a Gr¨obner-Shirshov basis, then one may construct a Gr¨obner-Shirshov basis R of Id(S) by adding at each step a non-trivial composition of previous poly-nomials (and reducing it by the ELW’s of previous polypoly-nomials and dividing by the leading coefficient). This process is called the Shirshov algorithm. In general, the Shirshov algorithm is infinite. The reader is referred to [5, 6, 7, 12, 13] for some recent work on Gr¨obner-Shirshov bases.
For a semigroup S, we denote by L and R the left and right Green’s congruences on S, respectively. Two elements a, b ∈ S are L-equivalent (aLb) if they generate the same left ideal, i.e., if S1a = S1b. In a similar way, two elements a, b ∈ S are R-equivalent (aRb) if aS1 = bS1. The intersection of L and R is denoted by H, while the smallest equivalence containing both L and R is denoted by D. Additionally, aL∗b if for all x, y ∈ S, ax = ay if and only if bx = by. Dually, aR∗b if for all x, y ∈ S, xa = ya if and only if xb = yb. Similarly to the relations D and H, the smallest equivalence containing both L∗ and R∗ is denoted by D∗ and the intersection of them is denoted by H∗. Evidently, L ⊆ L∗, R ⊆ R∗, and hence, D ⊆ D∗ and H ⊆ H∗. H∗ is an H∗-class in the semigroup S.
2 Generalized Bruck-Reilly ∗-Extension Let T be a monoid with H∗
1 (resp., H1) as the H∗-class (resp., H-class) containing the identity 1T of T . Assume that β and γ are morphisms from T into H∗
1. Let u be an element in H1 and let λu be the inner automorphism of H∗
1 defined by x 7→ uxu−1 such that γλu= βγ. Now we can consider S = N0× N0× T × N0× N0 into a semigroup by defining
(m, n, v, p, q)(m′, n′, v′, p′, q′) = ¡ m, n − p + max(p, n′), (vβmax(p,n′)−p )(v′βmax(p,n′)−n′ ), p′− n′+ max(p, n′), q′¢ if q = m′, ¡ m, n, v(((u−n′ (v′γ)up′ )γq−m′−1 )βp), p, q′− m′+ q¢ if q > m′, ¡ m − q + m′, n′, (((u−n(vγ)up)γm′−q−1 )βn′ )v′, p′, q′¢ if q < m′, where β0, γ0 are interpreted as the identity map of T and u0 is interpreted as the identity 1T of T . The monoid S = N0× N0× T × N0× N0 constructed above is called the generalized Bruck-Reilly ∗-extension of T determined by the morphisms β, γ and the element u. This monoid is denoted by S = GBR∗(T ; β, γ; u) and its identity is the element (0, 0, 1T, 0, 0).
Lemma 2.1. Suppose thatX is a generating set for the monoid T . Then
{(0, 0, x, 0, 0) : x ∈ X} ∪ (0, 1, 1T, 0, 0) ∪ (1, 0, 1T, 0, 0) ∪ (0, 0, 1T, 1, 0) ∪ (0, 0, 1T, 0, 1)
is a generating set for the monoidS = GBR∗(T ; β, γ; u).
Proof. For v, v1, v2∈ T and mi, ni, pi, qi∈ N0(1 ≤ i ≤ 2), we can easily show that the proof follows from the equations
(0, 0, v1, 0, 0)(0, 0, v2, 0, 0) = (0, 0, v1v2, 0, 0), (m1, 0, 1T, 0, 0)(m2, 0, 1T, 0, 0) = (m1+ m2, 0, 1T, 0, 0),
(0, n1, 1T, 0, 0)(0, n2, 1T, 0, 0) = (0, n1+ n2, 1T, 0, 0),
(0, 0, 1T, p1, 0)(0, 0, 1T, p2, 0) = (0, 0, 1T, p1+ p2, 0), (0, 0, 1T, 0, q1)(0, 0, 1T, 0, q2) = (0, 0, 1T, 0, q1+ q2), (m1, 0, 1T, 0, 0)(0, n2, 1T, 0, 0) = (m1, n2, 1T, 0, 0), (0, 0, 1T, p1, 0)(0, 0, 1T, 0, q2) = (0, 0, 1T, p1, q2), (m1, n1, 1T, 0, 0)(0, 0, v, 0, 0)(0, 0, 1T, p1, q1) = (m1, n1, v, p1, q1), as required. ¤
Theorem 2.2. Let T be a monoid defined by the presentation < X; R >, and
letβ, γ be morphisms from T into H∗
1. Then the monoidS = GBR∗(T ; β, γ; u) is
defined by the presentation
hX, y, z, a, b : R, (1) yz = 1, ba = 1, (2) yx = (xγ)y, xz = z(xγ), (3) bx = (xβ)b, xa = a(xβ), (4) yb = uy, ya = u−1y, (5) bz = zu, az = zu−1i, (6) wherex ∈ X.
Proof. Denote the set X ∪ {y, z, a, b} by Y . Let φ : Y∗→ GBR∗(T ; β, γ; u) be the homomorphism defined by
xφ = (0, 0, x, 0, 0), yφ = (0, 0, 1T, 0, 1), zφ = (1, 0, 1T, 0, 0), aφ = (0, 1, 1T, 0, 0), bφ = (0, 0, 1T, 1, 0).
By Lemma 2.1, φ is an epimorphism. Let us check whether GBR∗(T ; β, γ; u) satis-fies relations (1)–(6). Since the relations R hold on T , by Lemma 2.1, they also hold on GBR∗(T ; β, γ; u). The remaining relations (2)–(6) can be checked as follows:
(2) : (0, 0, 1T, 1, 0)(0, 1, 1T, 0, 0) = (0, 0, 1T, 0, 0), (0, 0, 1T, 0, 1)(1, 0, 1T, 0, 0) = (0, 0, 1T, 0, 0); (3) : (0, 0, 1T, 0, 1)(0, 0, x, 0, 0) = (0, 0, xγ, 0, 1) = (0, 0, xγ, 0, 0)(0, 0, 1T, 0, 1), (0, 0, x, 0, 0)(1, 0, 1T, 0, 0) = (1, 0, xγ, 0, 0) = (1, 0, 1T, 0, 0)(0, 0, xγ, 0, 0); (4) : (0, 0, 1T, 1, 0)(0, 0, x, 0, 0) = (0, 0, xβ, 1, 0) = (0, 0, xβ, 0, 0)(0, 0, 1T, 1, 0), (0, 0, x, 0, 0)(0, 1, 1T, 0, 0) = (0, 1, xβ, 0, 0) = (0, 1, 1T, 0, 0)(0, 0, xβ, 0, 0); (5) : (0, 0, 1T, 0, 1)(0, 0, 1T, 1, 0) = (0, 0, u, 0, 1) = (0, 0, u, 0, 0)(0, 0, 1T, 0, 1), (0, 0, 1T, 0, 1)(0, 1, 1T, 0, 0) = (0, 0, u−1, 0, 1) = (0, 0, u−1, 0, 0)(0, 0, 1T, 0, 1); (6) : (0, 0, 1T, 1, 0)(1, 0, 1T, 0, 0) = (1, 0, u, 0, 0) = (1, 0, 1T, 0, 0)(0, 0, u, 0, 0), (0, 1, 1T, 0, 0)(1, 0, 1T, 0, 0) = (1, 0, u−1, 0, 0) = (1, 0, 1T, 0, 0)(0, 0, u−1, 0, 0), where x ∈ X. Therefore, φ induces an epimorphism φ from the monoid M defined by (1)–(6) onto GBR∗(T ; β, γ; u).
For a given word w, let |w| denote the length of w. We now show that, in M , any non-empty word w ∈ Y∗ is equal to a word of the form f (m, n, v, p, q) =
zmanvbpyq, where m, n, p, q ∈ N0 and v ∈ X∗. Here, if |w| = 1, then we have w = x = f (0, 0, x, 0, 0) for some x ∈ X, or w = y = f (0, 0, 1T, 0, 1), or w = z = f (1, 0, 1T, 0, 0), or w = a = f (0, 1, 1T, 0, 0), or w = b = f (0, 0, 1T, 1, 0). Now let us suppose inductively that every word of length less than k can be reduced to a word of the form f (m, n, v, p, q) and let |w| = k. Then w can be written as f (m, n, v, p, q)x (x ∈ X), or f (m, n, v, p, q)y, or f (m, n, v, p, q)a, or f (m, n, v, p, q)b, or f (m, n, v, p, q)z. Now f (m, n, v, p, q)x = zmanvbpyqx = zmanv((xγq)βp)bpyq= f (m, n, v((xγq)βp), p, q); f (m, n, v, p, q)y = zmanvbpyqy = zmanvbpyq+1= f (m, n, v, p, q + 1); f (m, n, v, p, q)a = zmanvbpyqa = f (m, n, v((u−1γq−1)βp), p, q) if q ≥ 1, f (m, n, v, p − 1, 0) if q = 0, p ≥ 1, f (m, n + 1, vβ, 0, 0) if q = p = 0; f (m, n, v, p, q)b = zmanvbpyqb = ½ f (m, n, v((uγq−1)βp), p, q) if q ≥ 1, f (m, n, v, p + 1, 0) if q = 0; f (m, n, v, p, q)z = zmanvbpyqz = ½ f (m, n, v, p, q − 1) if q ≥ 1, f (m + 1, 0, u−n(vγ)up, 0, 0) if q = 0. So the inductive step is complete.
Finally, if (f (m1, n1, v1, p1, q1))φ = (f (m2, n2, v2, p2, q2))φ, then (zm1an1v1bp1yq1)φ = (zm2an2v2bp2yq2)φ,
and so (m1, n1, v1, p1, q1) = (m2, n2, v2, p2, q2), where v1, v2∈ X∗and mi, ni, pi, qi ∈ N0 (1 ≤ i ≤ 2). Hence, v1= v2 in T and m1= m2, n1= n2, p1= p2 and q1= q2. Since the presentation of GBR∗(T ; β, γ; u) contains R, we deduce that v1= v2holds in M , and thus zm1an1v1bp1yq1 = zm2an2v2bp2yq2 in M . Therefore, φ is injective,
as required. ¤
Corollary 2.3. Letv be an arbitrary word in X∗. The relations
ymv = (vγm)ym, vzm= zm(vγm), bnv = (vβn)bn, van= an(vβn), ymbn= (uγm−1)nym, yman= (u−1γm−1)nym,
bnzm= zm(uγm−1)n, anzm= zm(u−1γm−1)n
hold in GBR∗(T ; β, γ; u) for all m, n ∈ N0. As a consequence, every word w ∈ (X ∪ {y, z, a, b})∗is equal inGBR∗(T ; β, γ; u) to a word of the form zmanvbpyq for
somev ∈ X∗andm, n, p, q ∈ N0.
Now we discuss Gr¨obner-Shirshov bases of the generalized Bruck-Reilly ∗-exten-sion. Let T be the monoid defined by the presentation hX; Ri. By Theorem 2.2, the generalized Bruck-Reilly ∗-extension GBR∗(T ; β, γ; u) is defined by a presentation
hX, y, z, a, b : R, yz = 1, ba = 1, yx = (xγ)y, xz = z(xγ), bx = (xβ)b, xa = a(xβ), yb = uy, ya = u−1y, bz = zu, az = zu−1i, (7) where x ∈ X. We note that throughout this section we assume that |xγ| = |xβ| = |u| = 1. Hence, we get xγ, xβ, u ∈ X. Now let us order the set (X ∪ {y, z, a, b})∗
lexicographically by using y > b > x > a > z (x ∈ X), and then consider the polynomials
(i) r − v, (ii) yz − 1, (iii) ba − 1, (iv) yx − (xγ)y, (v) bx − (xβ)b, (vi) xz − z(xγ), (vii) xa − a(xβ), (viii) yb − uy, (ix) ya − u−1y, (x) az − zu−1, (xi) bz − zu,
where r = v ∈ R. Also, we can assume that R is a minimal Gr¨obner-Shirshov basis for the monoid T in a sense that the leading monomials are not contained in each other as subwords, in particular, they are pairwise different. Now let us label some words with the notations
r : word without the last letter of the word r, r : word without the first letter of the word r.
Theorem 2.4. A Gr¨obner-Shirshov basis for GBR∗(T ; β, γ; u) presented by (7)
consists of polynomials(i)–(xi).
Proof. We need to prove that all compositions of polynomials (i)–(xi) are trivial. To do that, firstly, let us consider intersection compositions of (i) with (vi) and (vii). The ambiguities are the following:
(i) ∧ (vi) : w = rxz,
(f, g)w = (r − v)z − r(xz − z(xγ)) = rz − vz − rz + rz(xγ) = rz(xγ) − vz = z(rγ)(xγ) − z(vγ) = z(rγ) − z(vγ) ≡ 0;
(i) ∧ (vii) : w = rxa,
(f, g)w = (r − v)a − r(xa − a(xβ)) = ra − va − ra + ra(xβ) = ra(xβ) − va = a(rβ)(xβ) − a(vβ) = a(rβ) − a(vβ) ≡ 0.
Next we proceed with intersection composition of (iii) with (x), and we have (iii) ∧ (x) : w = baz,
(f, g)w = (ba − 1)z − b(az − zu−1) = baz − z − baz + bzu−1 = bzu−1− z = zuu−1− z ≡ 0.
By intersection compositions of (iv) with (i), (vi) and (vii), we get the following ambiguities:
(iv) ∧ (i) : w = yxr,
(f, g)w = (yx − (xγ)y)r − y(r − v) = yxr − (xγ)yr − yr + yv = yv − (xγ)yr = (vγ)y − (xγ)(rγ)y = (vγ)y − (rγ)y ≡ 0; (iv) ∧ (vi) : w = yxz,
(f, g)w = (yx − (xγ)y)z − y(xz − z(xγ)) = yxz − (xγ)yz − yxz + yz(xγ) = −(xγ)yz + yz(xγ) ≡ 0;
(iv) ∧ (vii) : w = yxa,
(f, g)w = (yx − (xγ)y)a − y(xa − a(xβ)) = yxa − (xγ)ya − yxa + ya(xβ) = ya(xβ) − (xγ)ya = u−1y(xβ) − (xγ)u−1y
= u−1((xβ)γ)y − (xγ)u−1y,
and since γλu= βγ, we get u−1((xβ)γ)y − (xγ)u−1y = u−1((xγ)λu)y − (xγ)u−1y = u−1u(xγ)u−1y − (xγ)u−1y ≡ 0.
Now we consider intersection compositions of (v) with (i), (vi) and (vii): (v) ∧ (i) : w = bxr,
(f, g)w = (bx − (xβ)b)r − b(r − v) = bxr − (xβ)br − br + bv = bv − (xβ)br = (vβ)b − (xβ)(rβ)b = (vβ)b − (rβ)b ≡ 0;
(v) ∧ (vi) : w = bxz,
(f, g)w = (bx − (xβ)b)z − b(xz − z(xγ)) = bxz − (xβ)bz − bxz + bz(xγ) = bz(xγ) − (xβ)bz = zu(xγ) − (xβ)zu = zu(xγ) − z((xβ)γ)u, and since γλu = βγ, we obtain zu(xγ) − z((xβ)γ)u = zu(xγ) − z((xγ)λu)u = zu(xγ) − zu(xγ)u−1u ≡ 0.
(v) ∧ (vii) : w = bxa,
(f, g)w = (bx − (xβ)b)a − b(xa − a(xβ)) = bxa − (xβ)ba − bxa + ba(xβ) = ba(xβ) − (xβ)ba ≡ 0.
Next we proceed with intersection compositions of (vii) with (x), (viii) with (iii), (v) and (xi), and (ix) with (x), hence we get
(vii) ∧ (x) : w = xaz,
(f, g)w = (xa − a(xβ))z − x(az − zu−1) = xaz − a(xβ)z − xaz + xzu−1 = xzu−1− a(xβ)z = z(xγ)u−1− az((xβ)γ),
and since γλu = βγ, we get z(xγ)u−1− az((xβ)γ) = z(xγ)u−1− az((xγ)λu) = z(xγ)u−1− azu(xγ)u−1= z(xγ)u−1− zu−1u(xγ)u−1≡ 0.
(viii) ∧ (iii) : w = yba,
(f, g)w = (yb − uy)a − y(ba − 1) = yba − uya − yba + y = y − uya = y − uu−1y ≡ 0;
(viii) ∧ (v) : w = ybx,
(f, g)w = (yb − uy)x − y(bx − (xβ)b) = ybx − uyx − ybx + y(xβ)b = y(xβ)b − uyx = ((xβ)γ)yb − uyx = ((xβ)γ)uy − u(xγ)y, and we get ((xβ)γ)uy − u(xγ)y = ((xγ)λu)uy − u(xγ)y = u(xγ)u−1uy − u(xγ)y ≡ 0 since γλu= βγ.
(viii) ∧ (xi) : w = ybz,
(f, g)w = (yb − uy)z − y(bz − zu) = ybz − uyz − ybz + yzu = yzu − uyz ≡ 0;
(ix) ∧ (x) : w = yaz,
(f, g)w = (ya − u−1y)z − y(az − zu−1) = yaz − u−1yz − yaz + yzu−1 = yzu−1− u−1yz ≡ 0.
Finally, it remains to check compositions of inclusion of polynomials (i)–(xi). But it is clear that there are no compositions of this type.
Hence, the result follows. ¤
Acknowledgements. The authors thank the referee for his/her valuable suggestions and
remarks which improved the understanding and presenting of the paper. The authors would also like to express their deepest gratitude to Professor Bokut for his kind guidance and useful discussions.
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