 Open Access
 Total Downloads : 67
 Authors : Flamure Sadiki, Alit Ibraimi, Azir Jusufi
 Paper ID : IJERTV5IS080296
 Volume & Issue : Volume 05, Issue 08 (August 2016)
 Published (First Online): 24082016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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·)

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

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,

if u is right identity element in (B, ), than hold the equivalence
ab c q(a, b, u, c), a,b, c B.

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.

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:

AA DC D C ;

AB , AB AB ;

If a direction lines of two nonzero vectors (Fig. 1).
AB , DC
are distinct lines then,
AB DC
AB  DC and
AD  BC

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 :

A B C . In this case from

A B C . In this case from

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.

B A C . In this case from paralelograms;
AB DC
we have
AB DA ;by (iv), the point
D is determine from two

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= YEXY = 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= EASo, 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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