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

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 · and called multiplication in B. Groupoid (B, ·) is called right transitive groupoid (shortly Ward groupoid) if

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

Definition 1.3. Ward groupoid (B, ·) 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, ·) 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, ·)

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, ·) 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, · ), exactly of its ternarygroupoid of a given element.

   Theorem 1.2. Existence of a substractive groupoid (B, ·) 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, · ) 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

   AD || BC

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.

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