# Desargues Systems and a Model of a Laterally Commutative Heap in Desargues Affine Plane

DOI : 10.17577/IJERTV5IS080296

Text Only Version

#### Desargues Systems and a Model of a Laterally Commutative Heap in Desargues Affine Plane

Flamure Sadiki1 , University of Tetovo, Tetovo, Macedonia.

Alit Ibraimi2, University of Tetovo, Tetovo, Macedonia.

Azir Jusufi3 University of Tetovo, Tetovo, Macedonia.

Abstract – In this paper we have obtained some property of heaps like algebraic structure with a ternary operation combined whith groupoids, by putting concept of ternary operation from multiplication, heap from multiplication and ternary groupoid. Also we show that in the same set, the existence of a laterally commutative heap, existence of a parallelogram space, existence of a narrowed Desargues system and existence of a subtractive grupoid are equivalent to each other. So, we constructed a model of a laterally commutative heap in Desargues affine plane.

Key words: Heaps, ward groupoids, subtractive groupoids, parallelogram space, narrowed Desargues system, laterally commutative heap.

1. LATERALLY COMMUTATIVE HEAP AND SUBTRACTIVE GROUPID IN THE SAME SET In the terminology of [1] we define the following definitions.

Definition 1.1. Let

[ ]: B3 B be a ternary operation in a nonempty set B. (B, [ ]) is called heap if a,b,c,d,e B ,

[[abc]de] [ab[cde]] [abb] [bba] a

(1)

(2)

Definition 1.2. Let B2 B be a binary operation in a set B, denoted whith Â· and called multiplication in B. Groupoid (B, Â·) is called right transitive groupoid (shortly Ward groupoid) if

a,b, c B , ac bc ab . (3)

Definition 1.3. Ward groupoid (B, Â·) is called right solvable if a,b B the equation ax b has a solution.

Right solvable Ward groupoid is a quasigroup [3].

Lemma1.1. [2] If (B, Â·) is right solvable Ward groupoid, than !u B such that a B ,

a a u , (4)

a u a . (5)

Consequence 1.1. Right solvable Ward groupoid (B, Â·)

1. has a unique right identity element and so is the element u b b , where b is an element of B;

2. the proposition hold

ab u ba, a,b B.

(5)

Definition 1.4. Let (B, ) be a multiplicative group and o be a fixed element in the set B. Ternary operation [ ] in B, defined as

[abc] ab oc , a,b, c B , (6)

is called ternary operation from multiplication by the element o, whereas heap (B, [ ]), in which ternary operation [ ] is defined from (6), is called heap operation from multiplication by the element o.

Proposition 1.1. If (B, ) is right solvable Ward groupoid and u is right identity element, than the structure (B, [ ]), in which

[ ] is ternary operation from multiplication by u, is heap.

In the terminology of [3] now we have this:

Definition 1.5. Ward groupoid (B, ) is called subtractive groupoid , if a,b B ,

a ab b . (7)

According to this definition, it is obvious that

x ab is a solution of the equation ax b .

Proposition 1.2. If the groupoid (B, ) is subtractive, than it is right solvable Ward groupoid.

Proposition 1.3. If the structure (B, [ ] ) is heap from multiplication by right identity element u of groupoid (B, ), which is subtractive, than hold the equation:

[abc] a bc , a,b, c B . (8)

Definition 1.6. Let (B, [ ]) is a ternary structure and o is a fixed element of the set B. Groupoid (B, ) is called ternary groupoid according o, if a,b B ,

a b = [abo] . (9)

Proposition 1.4. If the structure (B, [ ]) is heap, than its ternary groupoid (B, ) according to o is right solvable Ward groupoid with a right identity element the given element o B .

Definition 1.7. Heap (B, [ ]) is called laterally commutative heap , if a,b, c B ,

[abc]=[cba] . (11)

Proposition1.5. If the groupoid (B, ) is subtractive, than the structure (B, [ ] ), in which the ternary operation [ ] is defined from (8), is laterally commutative heap.

Proposition 1.6. . If the groupoid (B, ) is subtractive , than the structure (B, [ ] ), in which [ ] is ternary operation of multiplication by right identity element of B, is laterally commutative heap.

Proposition 1.7. If the structure (B, [ ]) is commutative heap in lateral way, than its ternarygroupoid (B, ) according to the element o of B, is subtractive.

Theorem 1.1. Existence of a right solvable Ward groupoids (B, Â·) with right identity element u gives the existence of a heap (B, [ ]), exactly corrensponding heap from multiciplation according u; existence of a heap (B, [ ]) gives the existence of a right solvable Ward groupoids (B, Â· ), exactly of its ternarygroupoid of a given element.

Theorem 1.2. Existence of a substractive groupoid (B, Â·) gives exsistence of a laterally commutative heap, exactly heap of multiciplation (B, [ ]) according to right identity element u; existence of a laterally commutative heap (B, [ ]) gives existence of a substractive groupoid, exactly ternargroupoid (B, Â· ) according to an element o in B.

Shortly, Theorem 1.2, shows that, the existence of a laterally commutative heap is equivalent to the existence of a subtractive groupoid on the same set.

2. DESARGUES SYSTEMS AND PARALLELOGRAM SPACE

Let q be a quarternary relation in a nonempty set B, respectively q B4. The fact that (x, y, z, u)q we can denote as

q(x, y, z, u) for (x, y, z, u) B4.

Definition 2.1. [4] Pair (B, q), where q is a quarternary relation in B, is called Desargues system if the following propositions are true:

D1. x, y, a,b, c, d B, q(x, a,b, y) q(x, c, d, y) q(c, a,b, d );

D2. x, y, a,b, c, d B, q(b, a, x, y) q(d, c, x, y) q(b, a, c, d );

D3. (a,b, c) B3, !d B, q(a,b, c, d ).

Lemma. 2.1. [5] If (B, q) is Desargues system, than we have:

1. a,b B, q(a, a,b,b) q(a,b,b, a).

2. a,b, c, d B, q(a,b,c,d ) q(b,a,d ,c),

q(a,b,c,d ) q(d ,c,b,a).

(12)

(13)

Theorem 2.1. [6] Let B be a set in which is defined ternary operation [ ] and a quarternary relation q, such that the equivalence is valid

[abc] d q(a,b, c, d ), a,b, c, d B . (14)

In these conditions, (B, q) is Desargues system, if and only if (B, [ ]) is a heap.

Definition 2.2. System (B, q) is called narrowed Desargues system if it holds:

D4.

a,b, c, d B, q(a,b, c, d ) q(a, d, c,b).

Theorem 2.2. Let B be a set in which is defined ternary operation [ ] and a quartenary relation q, such that satisfy (14).

In these conditions, (B, q) is narrowed Desargues system, if and only if (B, [ ]) is laterally comutative heap.

Definition 2.3. Pair (B, p), where p is a quartenary relation in B, is called parallelogram space if the propositions are valid:

P1. a,b, c, d B,

P2. a,b, c, d B,

p(a,b, c, d ) p(a, c,b, d );

p(a,b, c, d ) p(c, d, a,b);

P3. a,b, c, d, e, f B, p(a,b, c, d ) p(c, d, e, f ) p(a,b, e, f );

P4. (a,b, c) B3,!d B, p(a,b, c, d ).

Theorem 2.3. [6] Let B be a set in which are defined quartenary operations p, q such that satisfy the equivalence

q(a,b, c, d ) p(a,b, d, c), a,b, c, d B . (15)

In these conditions, (B, p) is a parallelogram space, if and only if (B, q) is narrowed Desargues system.

Theorem 2.4. Let B be a set in which is defined ternary operation [ ] and a quarternary relation p, such that hold the relation

[abc] d p(a,b, d, c), a,b, c, d B.

In these conditions, (B, p) is parallelogram space if and only if (B, [ ]) is a laterally commutative heap.

(16)

Theorem 2.5. Let B be a set in which is defined multiplication and a quarterny relation q, such that equivalence is valid

a bc d q(a,b,c,d ), a,b, c, d B.

In these conditions, (B, ) is a substractive groupoid, if and only if (B, q) is narrowed Desargues system.

(17)

Consequence 2.1. Let B be a set in which is defined multiplication and a quarternay relation q, such that equivalence is valid

a bc d p(a,b,d,c), a,b, c, d B.

In these conditions, (B, ) is subtractive groupoid, if and only if (B, p) is parallelogram space.

From Theorems 2.2, 2.3, 2.4 dhe 2.5, it is obviously that:

Theorem 2.6. In the same set, the existence of a laterally commutative heap, existence of a parallelogram space, existence of a narrowed Desargues system and existence of a subtractive grupoid are equivalent to each other.

Following proposition gives sufficient condition of existence of Desargues system.

Proposition 2.1 Let B be a set in which is defined multiplication and a quarternay relation q, such that equivalence is valid

ab cd q(a,b, d, c), a,b, c, d B.

In these conditions,

1. if u is right identity element in (B, ), than hold the equivalence

ab c q(a, b, u, c), a,b, c B.

2. if (B, ) is Ward quasigroup, than (B, q) is Desargues system.

(18)

Consequence 2.2. Let B be a set in which is defined multiplication and a quarternay relation q, such that equivalence (18)

hold.

In these conditions, if (B, ) is subtractive groupoid, than (B, q) is Desargues system.

3. Model of a laterally commutative heap in Desargues affine plane

Let incidence structure =(, , ) be an Desargues affine plane.

In Desargues affine plane vector is defined like an ordered pair of points from . If this point is a pair

( A, B) of

distinct point

A, B and we denote

AB .

AB is called zero vector if

A B .

Equality of vectors is defined:

1. AA DC D C ;

2. AB , AB AB ;

3. If a direction lines of two nonzero vectors (Fig. 1).

AB , DC

are distinct lines then,

AB DC

AB || DC and

4. If a direction lines of two nonzero vectors

AB , DC

Fig. 1

are the same lines, then

AB DC , when there exixsts a vector

MN with direction line MN AB such that

AB MN

and MN DC

(Fig. 2).

From this definition hold

AB DC AD BC

Fig. 2

(19)

When the points

A, B,C

are nocolinear, the point D , by (iii), are the four vertex of a parallelogram

ABCD . (Fig. 1).

When the points

A, B,C

are collinear, we have the following cases for determine the point D :

1. A B C . In this case from

2. A B C . In this case from

3. A B C . In this case from

AB DC AB DC AB DC

we have we have

we have

AA DA ; by (i) D A. AA DC ; by (i) D C . AB DB ; by (ii) D A.

4. B A C . In this case from paralelograms;

AB DC

we have

AB DA ;by (iv), the point

D is determine from two

5. A, B,C are distict. In this case by (iv), the point D is determined whith two paralelograms.

So, (A, B,C) 3, !D ,

AB DC . Let we determine in ternary operation [ ]: 3 such that:

[ ABC] D AB DC , (A, B,C) 3 (20)

In this way we constructed ternary structure (, [ ]) in an Desargues affine plane. In the following proposition we prove that this is the model of an laterally commutative heap in such plane.

Proposition 3.1. Ternary structure (, [ ]) is a laterally commutative heap.

Proof. Let A, B,C, D, E . We denote [ ABC] X , [ XDE] Y , [CDE] Z , [ ABZ ] T . By (19) and (20) we have:

[ABC]=X AB= XC;

(19) ;

[XDE]=Y XD= YE

XY = DE

AB

TZ;

(19)

[CDE]=Z CD= ZE

[ABZ ]=T AB= TZ

(19)

CZ = DE;

XY = CZ

XC

YZ

Hence, YZ =

TZ Y T [[ABC]DE] [AB[CDE]]. So,

[[ABC]DE] [AB[CDE]] , A, B,C, D, E ( )

In Fig. 3 we illustrate this.

Hence, by (20), [ ABB] D

AB DB . By (ii),

Fig. 3

AB DB D A . This implicate [ ABB] A . Also by (20),

[BBA] D BB DA . By (i), BB DA D A. This implicate [BBA] A . So hold,

[ABB] [BBA] A

( )

Finaly, for each three points

A, B,C

from , we have

[ABC] D AB= DC;

[CBA]=E CB= EA

So, we have

EA=CB

(19)

EC=DC D E. EC= AB

[ABC] [CBA],

(A, B,C) 3 ( )

The results ( ), ( ), ( ), by the Definition 1.1. and Definition 1.7, shows that this proposition hold.

REFERENCES

1. V.V.Vagner, Teorija obobenih grud i obobenih grupp, Mat. Sbornik 32(74)(1953), 545-632.

2. V.Volenec, Heaps and Right Solvable Ward groupoids, Journal of Algebra 156(1993), 1-4.

3. M. Polonijo, A note on Ward quasigroups. An Stiint. Univ. Al. I. Cuza Iasi. Sect.I a Mat. (N.S) 32 (1986), 5-10.

4. D.Vakarelov, Dezargovi sistemi, Godishnik Univ. Sofija Mat. Fak. 64 (1969-70), 227-235 (1971).

5. M. Polonijo, Desargues systems and Ward quasigroups, Rad Jugoslav. akad. znan.- umjet. mat. [444] (8) (1989), 97-102.

6. Z.Kolar, Heap-ternary algebraic structure, Mathematical Communications 5(2000), 87-95.