b b International Journal of Mathematics and M Computer Science, 15(2020), no. 2, 671–682 CS

Congruences and on n-ary

Chitlada Somsup, Utsanee Leerawat

Department of Mathematics Faculty of Science Kasetsart University Bangkok 10900, Thailand

email: [email protected], [email protected] (Received December 26, 2019, Accepted February 8, 2020)

Abstract In this paper, we introduce the concept of congruences on an n- ary semigroups and investigate their related properties. Moreover, we introduce quotient n-ary semigroups via congruence relations. Fur- thermore, we establish fundamental theorem of for n-ary and related properties with respect to congruence relations.

1 Introduction

An n-ary system, a generalization of algebraic structures which has many applications in different branches of mathematics, was initiated by Kasner [4] in 1904. For example, in the theory of automata (see [5]) n-ary semigroup and n-ary groups are used, some other n-ary systems are applied in the theory of quantum groups (see [3]) and combinatorics (see [18], [19]). Different applications of ternary structures in physics are described in Kerner [7]. In physics, such structures as n-ary Filippov algebras (see [1]) and n-Lie algebras (see [6]) are also used. Some n-ary structures induced by hypercubes have application in error-correcting and error-detecting coding theory, cryptology, as well as in the theory of (t, m, s)-nets (see [2]). Key words and phrases: n-ary semigroups, n-ary operation, congruence, homomorphism, . AMS (MOS) Subject Classifications: 08A02, 17A42, 08A30, 20N15, 16Y99. Corresponding author email: [email protected] ISSN 1814-0432, 2020, http://ijmcs.future-in-tech.net 672 C. Somsup, U. Leerawat

In 1928, W. D¨ornte introduced the notion of an n-ary , which is a generalization of the notion of a group to a nonempty G with an n-ary operation instead of a . By an n-ary operation is meant any mapping f : Gn → G from the n-th Cartesian power of G to G. The important study of n-ary groups and n-ary semigroups was done by Dudek (see [10]-[17]). The notion of congruence was introduced by Karl Fredrich Gauss in the beginning of the nineteenth century. It is well known that congruences always play an important role in the study of algebraic structures. In particular, congruences are a special type of equivalence relations that play a vital role in the study of quotient structures of different algebraic structures. In 1997, V. N. Dixit and S. Dewan [9] represented the concept of congruences on a ternary semigroups and they studied some interesting properties of them. In 2007, S. Kar and B. K. Maity [8] introduced some concepts such as cancellation congruences, group congruences and Rees congruences and investigated these congruences in ternary semigroups. In this paper, we introduce the concept of congruences on an n-ary semi- groups and investigate their related properties. Moreover, we establish the quotient n-ary semigroups via congruences. Furthermore, some homomor- phisms and related properties with respect to congruence relations are pro- posed.

2 Preliminaries

Throughout this paper, n is a positive greater than one. In this section, we start with some elementary notions that will be used in the sequel.

Definition 2.1. A nonempty set G is called an n-ary semigroup if there ex- n ists an n-ary operation G → G, written as (x1, x2,...,xn) 7→ x1x2 ··· xn, satisfying the following condition: n+i−1 x1x2 ··· xi−1 xk xn+ixn+i+1 ··· x2n−1  k=i  Q n+j−1 = x1x2 ··· xj−1 xk xn+jxn+j+1 ··· x2n−1, k=j ! Q for all x1, x2,...,x2n−1 ∈ G and 1 ≤ i < j ≤ n. Congruences and homomorphisms on n-ary semigroups 673

Remark 2.2 Let G be an ordinary semigroup under the binary operation

(x1, x2) 7→ x1 ∗ x2.

Then G, with the n-ary operation

(x1, x2,...,xn) 7→ x1 ∗ x2 ∗···∗ ((xi ∗ xi+1) ∗ xi+2) ∗···∗ xn, is an n-ary semigroup.

Example 2.3 (1) Let G = {−i, 0, i} be a of complex numbers. Then G is a 3-ary semigroup under the usual . However, G is not a 4-ary semigroup under the usual multiplication, because (i, −i, i, i)= −1 ∈/ G. (2) Let G = {2, 4, 6,...} be the set of all positive even numbers. Then G is an n-ary semigroup under the usual multiplication.

Definition 2.4 A nonempty set S of an n-ary semigroup G is called an n n-ary subsemigroup if xi ∈ S for all x1, x2,...,xn ∈ S. i=1 Q Example 2.5 The set of all , Z, forms an n-ary semigroup under the operation x1x2 ...xn = min{x1, x2,...,xn} for all x1, x2,...,xn ∈ Z. As a result, the set N = {1, 2, 3,...} of all natural numbers gives an n-ary subsemigroup.

Theorem 2.6 The nonempty intersection of any two n-ary subsemigroups of an n-ary semigroup G is an n-ary subsemigroup of G.

Proof. The proof is straightforward. The following corollary follows by induction:

Corollary 2.7 The nonempty intersection of any family of n-ary subsemi- groups of an n-ary semigroup G is an n-ary subsemigroup.

Definition 2.8 Let G1 and G2 be two n-ary semigroups. The mapping f : G1 → G2 is called (an n-ary semigroup) homomorphism from G1 into G2 if f(a1a2 ...an) = f(a1)f(a2) ...f(an), 674 C. Somsup, U. Leerawat

for all a1, a2,...,an ∈ G1.

A homomorphism f : G1 → G2 is called an isomorphism if it is both one-one and onto and in this case we say that the n-ary semigroups G1 and G2 are isomorphic and write G1 ∼= G2.

Theorem 2.9 Let G1,G2 and G3 be any n-ary semigroups and the map g1 : G1 → G2, g2 : G2 → G3 be homomorphisms. Then g2 ◦g1 is a homomorphism from G1 into G3.

Proof. The proof is straightforward.

3 Congruence relations and homomorphism

In this section, we introduce the concept of congruence relations and es- tablish quotient n-ary semigroups via congruence relations. Moreover, some homomorphisms and related properties with respect to congruence relations are provided.

Definition 3.1 An ρ on an n-ary semigroup G is said to be 1. a left congruence relation if (a, b) ∈ ρ implies

n−1 n−1

ti a, ti b ∈ ρ, i=1 ! i=1 ! ! Y Y for all t1,...,tn−1, a, b ∈ G.

2. an intra congruence relation if (a, b) ∈ ρ implies

(t1at2 ...tn−1, t1bt2 ...tn−1) ∈ ρ,

(t1t2at3 ...tn−1, t1t2bt3 ...tn−1) ∈ ρ, . .

(t1t2 ...tn−2atn−1, t1t2 ...tn−2btn−1) ∈ ρ, for all t1,...,tn−1, a, b ∈ G. Congruences and homomorphisms on n-ary semigroups 675

3. a right congruence relation if (a, b) ∈ ρ implies

n−1 n−1

a ti , b ti ∈ ρ, i=1 ! i=1 !! Y Y for all t1,...,tn−1, a, b ∈ G.

4. a congruence relation if (a1, b1),..., (an, bn) ∈ ρ implies

(a1 ··· an, b1 ··· bn) ∈ ρ, for all a1,...,an, b1,...,bn ∈ G.

Proposition 3.2 An equivalence relation ρ on an n-ary semigroup G is a congruence if and only if it is a left, an intra and a right congruence relations on G.

Proof. Assume that ρ is a congruence relation on G. Let a, b ∈ G be such that (a, b) ∈ ρ and let t1,...,tn−1 ∈ G. Since ρ is an equivalence relation on G, we have (t1, t1),..., (tn−1, tn−1) ∈ ρ. Then (t1 ...tn−1a, t1 ...tn−1b) ∈ ρ. So ρ is a left congruence relation on G. Similarly, ρ is an intra and a right congruence relation on G. Conversely, assume that ρ is a left, an intra and a right congruence rela- tion on G. Let a1,...,an, b1, b2,..., bn ∈ G be such that (a1, b1),..., (an, bn) ∈ ρ. Since ρ is a right congruence relation, we have (a1a2 ··· an, b1a2 ··· an) ∈ ρ. Since ρ is an intra congruence relation, we have

(b1a2a3 ··· an, b1b2a3 ··· an) ∈ ρ,

(b1b2a3a4 ··· an, b1b2b3a4 ··· an) ∈ ρ,

(b1b2b3a4a5 ··· an, b1b2b3b4a5 ··· an) ∈ ρ, . .

(b1 ··· bn−2an−1an, b1 ··· bn−2bn−1an) ∈ ρ.

Since ρ is a left congruence relation, we have (b1 ··· bn−1an, b1 ··· bn−1bn) ∈ ρ. By , (a1 ··· an, b1 ··· bn) ∈ ρ. Hence ρ is a congruence relation on G. 676 C. Somsup, U. Leerawat

Proposition 3.3 Let ρ be a congruence relation on an n-ary semigroup G. Then ρ ◦ ρ is a congruence relation on G.

Proof. Clearly, ρ ◦ ρ is a congruence relation on G. Let a1, a2,...,an, b1, b2,...,bn ∈ G be such that (a1, b1), (a2, b2),..., (an, bn) ∈ ρ ◦ ρ. Then there are c1,c2,...,cn ∈ G such that

(a1,c1) ∈ ρ and (c1, b1) ∈ ρ,

(a2,c2) ∈ ρ and (c2, b2) ∈ ρ, . .

(an,cn) ∈ ρ and (cn, bn) ∈ ρ.

Since ρ is a congruence relation, we have (a1a2 ··· an, c1c2 ··· cn) ∈ ρ and (c1c2 ··· cn, b1b2 ··· bn) ∈ ρ. Thus (a1a2 ··· an, b1b2 ··· bn) ∈ ρ ◦ ρ. Hence ρ ◦ ρ is a congruence relation on G.

Next, we will consider the relation between congruence relations and homomorphisms on n-ary semigroups.

Theorem 3.4 Let G1 and G2 be two n-ary semigroups and f : G1 → G2 be a homomorphism. Let ρ be a congruence relation on G1. Then

f(ρ)= {(f(x), f(y)) ∈ G2 × G2 | (x, y) ∈ ρ} is a congruence relation on G2.

Proof. Clearly, f(ρ) is an equivalence relation. Let x1, x2,...,xn, y1,y2,...,yn ∈ G1 be such that

(f(x1), f( y1)), (f(x2), f(y2)),..., (f(xn), f(yn)) ∈ f(ρ).

Then (x1,y1), (x2,y2),..., (xn,yn) ∈ ρ. Since ρ is a congruence relation, we have (x1x2 ··· xn, y1y2 ··· yn) ∈ ρ. So (f(x1x2 ··· xn), f(y1y2 ··· yn)) ∈ f(ρ). Since f is a homomorphism, (f(x1)f(x2) ··· f(xn), f(y1)f( y2) ··· f(yn)) ∈ f(ρ). Thus f(ρ) is a congruence relation on G2. Congruences and homomorphisms on n-ary semigroups 677

Definition 3.5 Let ρ be a congruence relation on an n-ary semigroup G. The of a ∈ G is defined to be the set

[a]ρ = {b ∈ G|(a, b) ∈ ρ}. G The quotient set /ρ is the set of all equivalence class of G with respect to G ρ; That is, /ρ = {[a]ρ | a ∈ G}.

Note that, for any a, b ∈ G, we have (a, b) ∈ ρ if and only if [a]ρ = [b]ρ.

Theorem 3.6 Let G be an n-ary semigroup and let ρ be a congruence G relation on G. Define multiplicative operation on /ρ by [a1]ρ[a2]ρ ··· [an]ρ = G G [a1a2 ...an]ρ for all [a1]ρ, [a2]ρ,..., [an]ρ ∈ /ρ. Then /ρ is an n-ary semi- group. We call it the quotient n-ary semigroup.

Proof. We first show that the multiplication operation is well-defined. Let a1,...,an, b1,...,bn ∈ G be such that [a1]ρ = [b1]ρ,..., [an]ρ = [bn]ρ. Then (a1, b1),..., (an, bn) ∈ ρ. Since ρ is a congruence relation, we have (a1 ··· an, b1 ··· bn) ∈ ρ. Thus [a1a2 ...an]ρ = [b1b2 ...bn]ρ. Therefore, the multiplicative operation is well-defined. G Next we will show that /ρ is an n-ary semigroup. Let a1,...,an, an+1,...,a2n−1 ∈ G. Then

[a1]ρ[a2]ρ ··· [an]ρ [an+1]ρ ... [a2n−1]ρ = [a1 ...an]ρ[an+1]ρ ... [a2n−1]ρ

  = [(a1 ...an)an+1 ...a2n−1]ρ

= [a1(a2 ...an+1)an+2 ...a2n−1]ρ

= [a1]ρ[a2 ...an+1]ρ[an+2]ρ ... [a2n−1]ρ n+1 2n−1

= [a1]ρ [ai]ρ [aj]ρ. i=2 ! j=n+2 Y Y n+1 2n−1 n−1 2n−1 Similarly, [a1]ρ [ai]ρ [aj]ρ = ··· = [ai]ρ [aj]ρ .  i=2  j=n+2 i=1 j=n ! G Q Q Q Q Thus /ρ is an n-ary semigroup.

Theorem 3.7 Let G be an n-ary semigroup and ρ be a congruence relation G on G. Then the mapping π : G → /ρ given by π(a) = [a]ρ is a surjective G homomorphism, called the natural homomorphism from G onto /ρ. 678 C. Somsup, U. Leerawat

Proof. For all a1,...,an ∈ G, we have

π(a1a2 ...an) = [a1 ...an]ρ = [a1]ρ[a2]ρ ... [an]ρ = π(a1) π(a2) ...π(an).

G Thus π is a homomorphism. Next, let [a]ρ ∈ /ρ, then π(a) = [a]ρ, which shows that π is surjective.

Definition 3.8 Let G1 and G2 be two n-ary semigroups and f : G1 → G2 be a mapping. Define a relation ker f on G1 by the rule

(a, b) ∈ ker f if and only if f(a)= f(b).

Theorem 3.9 Let G1 and G2 be two n-ary semigroups and the map f : G1 → G2 be a homomorphism. Then ker f is a congruence relation on G1.

Proof. First, we have to show ker f is an equivalence relation on G1. Let a, b, c ∈ G1. Then (a, a) ∈ ker f. Hence ker f is reflexive. If (a, b) ∈ ker f, then f(a) = f(b). So f(b) = f(a), that is (b, a) ∈ ker f. Thus ker f is symmetric. Suppose that (a, b) ∈ ker f and (b, c) ∈ ker f. Then f(a) = f(b) and f(b)= f(c). So f(a)= f(c), that is (a, c) ∈ ker f. Thus ker f is transitive. Therefore ker f is an equivalence relation. Next, we will show that ker f is a congruence. Let

(a1, b1), (a2, b2),..., (an, bn) ∈ ker f.

Then f(a1)= f(b1), f(a2)= f(b2), ...,f(an)= f(bn). So

f(a1a2 ...an)= f(a1)f(a2) ...f(an)= f(b1)f(b2) ...f(bn)= f(b1b2 ...bn).

Thus (a1a2 ...an, b1b2 ...bn) ∈ ker f. Hence ker f is a congruence on G1.

Theorem 3.10 (The Fundamental Theorem of Homomorphism for n-ary semigroups) Let G1 and G2 be two n-ary semigroups and the map f : G1 → G2 be a homomorphism. Then f(G1) is an n-ary subsemigroup of G2 and G1 ∼ /ker f = f(G1). Congruences and homomorphisms on n-ary semigroups 679

Proof. Let x1, x2,...,xn ∈ f(G1). Then there exist a1, a2,...,an ∈ G1 such that x1 = f(a1), x2 = f(a2),...,xn = f(an). So

x1, x2,...,xn = f(a1)f(a2) ...f(an)= f(a1a2 ...an) ∈ f(G1).

Thus f(G1) is an n-ary subsemigroup of G2. G1 Define α : /ker f → f(G1) by α ([a]ker f ) = f(a), for all a ∈ G1. Let a, b ∈ G1. [a]ker f = [b]ker f ⇔ (a, b) ∈ ker f ⇔ f(a) = f(b) ⇔ α([a]ker f ) = α([b]ker f )). Hence α is well-defined and one-to one. For any y ∈ f(G1). Then y = f(x) for some x ∈ G1. We have y = f(x)= α([x]ker f ), and so α is onto. G Finally, let [a1]ker f ,..., [an]ker f ∈ /ker f. Then α([a1]ker f ··· [an]ker f ) = α([a1 ··· an]ker f )= f(a1 ··· an)= f(a1) ··· f(an)= α([a1]ker f ) ··· α([an]ker f )). G1 ∼ Therefore, α is an isomorphism and /ker f = f(G1).

Theorem 3.11 Let G be an n-ary semigroup and f : G → G be a homo- morphism. If ρ is a congruence relation on G such that ρ ⊆ ker f, then there G G is a surjective homomorphism g : /ρ → /ker f such that g ◦ π1 = π2 where G G π1 : G → /ρ and π2 : G → /ker f are natural homomorphisms.

G G Proof. Define g : /ρ → /ker f by g([a]ρ) = [a]ker f for all a ∈ G. Let a, b ∈ G. Suppose that [a]ρ = [b]ρ. Then (a, b) ∈ ρ. Since ρ ∈ ker f, we have (a, b) ∈ ker f. Thus [a]ker f = [b]ker f , that is g([a]ρ) = g([b]ρ). Therefore, g G is well-defined. For any y ∈ /ker f, y = [a]ker f for some a ∈ G. We have G y = [a]ker f = g([a]ρ), and so g is onto. Next, let [a1]ρ,..., [an]ρ ∈ /ρ. Then

g([a1]ρ ··· [an]ρ)= g([a1 ··· an]ρ) = [a1 ··· an]ker f =

[a1]ker f ··· [an]ker f = g([a1]ρ) ··· g([an]ρ).

Thus g is a surjective homomorphism. Finally, let a ∈ G. Then g ◦ π1(a) = g(π1(a)) = g([a]ρ) = [a]ker f = π2(a). Consequently, g ◦ π1 = π2.

Theorem 3.12 Let G be an n-ary semigroup and let ρ and σ be congruence relations on G such that ρ ⊆ σ. Then there exists a surjective homomorphism G G G G g : /ρ → /σ such that g ◦ π1 = π2, where π1 : G → /ρ and π2 : G → /σ are natural homomorphisms. 680 C. Somsup, U. Leerawat

Proof. The proof is similar to that of Theorem 3.11.

As an immediate consequence of Theorems 3.12 and 2.9, we have the following:

Corollary 3.13 Let G be an n-ary semigroup and σ1, σ2,...,σn(n > 2) be congruence relations on G such that σ1 ⊆ σ2 ⊆ ... ⊆ σn. Then there exists a surjective homomorphism g : G/ → G/ . σ1 σn

Theorem 3.14 Let ρ and σ be congruence relations on an n-ary semigroup σ G G G such that ρ ⊆ σ. Then /ρ = ([a]ρ, [b]ρ) ∈ /ρ × /ρ | (a, b) ∈ σ is a Gn o congruence relation on G/ and /ρ σ ∼= G/ . ρ /ρ σ   .  σ G Proof. First, we have to show /ρ is an equivalence relation on /ρ. Let G σ σ [a]ρ, [b]ρ, [c]ρ ∈ /ρ. Clearly, ([a]ρ, [a]ρ) ∈ /ρ. So /ρ is reflexive. If ([a]ρ, [b]ρ) ∈ σ /ρ, then (a, b) ∈ σ. Since σ is symmetric, we have (b, a) ∈ σ. Thus ([b]ρ, [a]ρ) ∈ σ σ σ /ρ, and so /ρ is symmetric. Suppose that ([a]ρ, [b]ρ), ([b]ρ, [c]ρ) ∈ /ρ. Then (a, b) ∈ σ and (b, c) ∈ σ. Since σ is transitive, we have (a, c) ∈ σ. Thus σ σ σ ([a]ρ, [c]ρ) ∈ /ρ, and so /ρ is transitive. Therefore /ρ is an equivalence relation. σ G Next, we will show that /ρ is a congruence relation on /ρ. Let ([a1]ρ, [b1]ρ), σ ([a2]ρ, [b2]ρ),..., ([an]ρ, [bn]ρ) ∈ /ρ. Then (a1, b1), (a2, b2),... (an, bn) ∈ σ. Since σ is a congruence relation on G, (a1a2 ...an, b1b2 ...bn) ∈ σ. This σ σ implies that ([a1a2 ...an]ρ, [b1b2 ...bn]ρ) ∈ /ρ, and hence /ρ is a congruence relation. G G Finally, let g : /ρ → /σ be a mapping defined by g([a]ρ)= πσ(a), for all G σ a ∈ G, where πσ : G → /σ is a natural homomorphism. Let ([a]ρ, [b]ρ) ∈ /ρ. Then (a, b) ∈ σ ⇔ [a]ρ = [b]ρ ⇔ πσ(a) = πσ(b) ⇔ g([a]ρ) = g([b]ρ) ⇔ ([a]ρ, [b]ρ) ∈ σ G/ ∼ G ker g. Thus ker g = /ρ. By Theorem 3.10, we have ρ ker g = g /ρ = G  .   G/ . Since ker g = σ/ , /ρ σ ∼= G/ . σ ρ /ρ σ   . 

Corollary 3.15 Let G be an n-ary semigroup and let σ1, σ2,...,σn be congruence relations on G such that σ1 ⊆ σ2 ⊆ ... ⊆ σn. Then for each Congruences and homomorphisms on n-ary semigroups 681 i =1,...,n − 1,

σi+1/ = ([a] , [b] ) ∈ G/ × G/ | (a, b) ∈ σ +1 is a congruence relation σi σi σi σi σi i on G/ andn o σi G/ ∼ G σi σi+1/ = /σ +1. σi i  . Moreover, for each i =1,...,n −2, the mapping

G/ G/ ϕi : σi σi+1 → σi+1 σi+2 . /σ /σ +1   i   i .  .  is a surjective homomorphism.

Proof. This follows from Theorems 3.14 and 3.7.

4 Acknowledgement

This research is supported by the Faculty of Science, Kasetsart University, Bangkok, Thailand (International SciKU Branding, ISB).

References

[1] A. P. Pojidaev, Enveloping algebras of Fillipov algebras, Comm. Alge- bra, 31, (2003), 883–900.

[2] C. F. Laywine, G. L. Mullen, Discrete Mathematics Using Latin Squares, Wiley, New York, 1998.

[3] D. Nikshych, L. Vainerman, Finite Quantum Groupoids and Their Ap- plications, New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., 43, (2002), 211–216.

[4] E. Kasner, An extension of the group concept, Bull. Amer. Math. Soc. 10, (1904), 290-291.

[5] J. W. Grzymala-Busse, Automorphisms of polyadic automata, J. Assoc. Comput. Mach., 16, (1969), 208–219. 682 C. Somsup, U. Leerawat

[6] L. Vainerman, R. Kerner, On special classes of n-algebras, J. Math. Phys., 37, (1996),2553–2565.

[7] R. Kerner, Ternary Algebraic Structures and Their Applications in Physics. Univ. P. and M. Curie, Paris, 2000.

[8] S. Kar and K. Maity . Congruences on ternary semigroups, J. Chungcheong Math. Soc., 20, (2007), 191–201.

[9] V. N. Dixit, S. Dewan, Congruence and Green’s equivalence relation on ternary semigroup, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 46, (1997), 103–117.

[10] W. A. Dudek, Remarks on n-groups, Demonstratio Math., 13, (1980), 165–181.

[11] W. A. Dudek, Autodistributive n-groups, Comment. Math. Ann. Soc. Math. Polon. Prace Mat., 23, (1983), 1–11.

[12] W. A. Dudek, On (i, j)-associative n-groupoids with the non-empty cen- ter, Ricer. Mat., (Napoli), 35, (1986), 105–111.

[13] W. A. Dudek . Idempotents in n-ary semigroups, Southeast Asian Bull. Math., 25, (2001), 97–104.

[14] W. A. Dudek, Remarks to Glazek’s results on n-ary groups, Disc. Math. Gen. Alg. Appl., 27, (2007), 199–233.

[15] W. A. Dudek, I. Grozdinska, On ideals in regular n-semigroups, Mat. Bilten Skopje, 4, (1980), 25–44.

[16] W. A. Dudek, J. Michalski, On retracts of polyadic groups, Demonstra- tio Math., 15, (1982), 783–805.

[17] W. D¨ornte, Unterschungen ¨uber einen verallgemeinerten Gruppenbe- griff, Math. Z., 29, (1929), 1–19.

[18] Z. Stojakovic, W. A. Dudek, Single identities for varieties equivalent to quadruple systems, Discrete Math., 183, (1998), 277-284.

[19] Z. Stojakovic, W. A. Dudek , Conjugate quasigroups, Quasi- groups and Related Systems, 13, (2005), 157–174.