units in quasigroups · introduction problems 1st family { identities similar to weak associativity...
TRANSCRIPT
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Units in quasigroups
Aleksandar Krapez(with J. Fempl–Madjarevic and V. A. Shcherbacov)
Mathematical Institute of the SASABelgrade, Serbia
AAA 94NSAC 2017
Novi Sad, June 14–18, 2017
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Dedication
Dedicated to B. Seselja,
colleague and friend
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Quasigroups
Quasigroups: the first definition
A quasigroup is a groupoid (Q; ·) such that linear equations:
a · x = b y · a = b
are uniquely solvable for all a, b ∈ Q.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Quasigroups
Quasigroups: the second definition
Quasigroups are algebras (Q; ·, /, \) satisfying:
xy/y = x x\xy = y(x/y)y = x x(x\y) = y
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Quasigroups
Sometimes they are called
equasigroups
primitive quasigroups
3–quasigroups
A class of all 3–quasigroups is a variety.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Quasigroups
The easiest way to ’understand’ quasigroups is:
Quasigroups are ’not necessarily associative’ groups. (WRONG!)
Groups are associative quasigroups. (right)
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Quasigroups
Another ’deficiency’ of quasigroupsis that they do not necessarily have a unit.
unit = our name for neutral element or identity element.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Units
unit symbol id 1 id 3
none (Q) x = x x = xleft (eQ) ex = x x/x = y/y
right (Qe) xe = x x\x = y\ymiddle (U) xx = e xx = yy` + r (1) ex = x , xe = x x/x = y\y` + m (eU) ex = x , xx = e x/x = yyr + m (Ue) xe = x , xx = e x\x = yy
` + r + m (U1) ex = x, xe = x, xx = e x/x = y\y = zz
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Units
� ��Q
� ������eQ � ��
U � ��QQQQ
Qe
� ��������
eU � ��������
SSSSSS
1 � ��SSSSSS
Ue
� ������QQQQ
U1
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Belousov’s problem
Problem (Belousov (1969))
How to recognize identities which force quasigroups satisfyingthem to be loops?
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Examples
Associativity:x · yz = xy · z
which defines groups.
Four Moufang identities:
z(x · zy) = (zx · z)y
x(z · yz) = (xz · y)z
zx · yz = (z · xy)z
zx · yz = z(xy · z)
which define Moufang loops (Kunen (1996), using Prover).Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Another problem
Problem (Krapez, Shcherbacov)
How to recognize identities which force quasigroups satisfyingthem to have {left, right, middle} unit?
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Examples
Left Bol identity:x(y · xz) = (x · yx)z
implies existence of a right unit.
Right Bol identity:x(yz · y) = (xy · z)y
implies existence of a left unit.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Weak associativity
J. D. H. Smith (2007) proved that the identity of weakassociativity:
x(y/y) · z = x · (y/y)z
implies (1).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Identites similar to weak associativity
Theorem (Krapez, Shcherbacov)• Identity x(y\y) · z = x · (y\y)z implies (1).
• Identity (x\yy)\z = x\(yy\z) implies (eU).• Identity (x\(y/y))\z = x\((y/y)\z) implies (eU).
• Identity (x/yy)/z = x/(yy/z) implies (Ue).• Identity (x/(y\y))/z = x/((y\y)/z) implies (Ue).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Weak associativity – general form
Theorem (Krapez)
Let (Q;A) be cancellative groupoid, (Q;B) groupoid, (Q;C ) rightquasigroup and (Q;D) left quasigroup. If they satisfy the identity:
xA((yBy)Cz) = (xD−1(yDy))Az (WA)
then there are e, i ∈ Q such that:
e is the middle unit for B,
e is the left unit for C ,
i is the middle unit for D.
The converse also holds.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Parastrophes
Operations ·, ∗, /, \, //, \\ are parastrophes of · (and of each other):
x · y = z iff x\z = y iff z/y = x iffy ∗ x = z iff z\\x = y iff y//z = x
There are six parastrophes for any quasigroup.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Weak associativity for parastrophes
We are interested in (WA) when A,B,C ,D are parastrophes of aquasigroup ·. Then:
All four operations are quasigroups;
The units e and i are equal;
e = i is some kind of unit for ·.
On the contrary,the choice of A does not impose any restriction on ·.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Partitions of Π
Let us define two partitions on Π = {·, ∗, /, \, //, \\}:
First: L = {/, //},R = {\, \\},M = {·, ∗}
Second: ` = {·, \}, r = {∗, //},m = {/, \\}.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Classification of WA–identities
Theorem (Krapez)
(WA) ⇔ (eQ) iff L`Liff A ∈ Π,B ∈ L,C ∈ `,D ∈ L.
There are 48 WA–identities equivalent to (eQ).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Example
Let A be \, B be //, C be · and D be /.Then D−1 is \\ and the appropriate WA–identity is:
x\((y//y) · z) = (x\\(y/y))\z
Translating to the language of quasigroups, we getthat all quasigroups satisfying:
x\((y/y) · z) = ((y/y)\x)\z
have left unit.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Classification of WA–identities
Analogously:
There are 48 WA–identities equivalent to (Qe).
There are 48 WA–identities equivalent to (U).
There are 288 WA–identities equivalent to (1).
There are 288 WA–identities equivalent to (eU).
There are 288 WA–identities equivalent to (Ue).
There are 288 WA–identities equivalent to (U1).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Closed identities
Definition
Let s = t be a quasigroup identity.A subterm u of s (t) is closed if
u is a variable, or
for every variable x from s = t,u either contains all appearances of x or none.
Identity s = t is closed if every subterm u of s (t) is closed.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Examples
1. Trivial identity x = x is closed.
2. The identity xy = xy is trivial, but not closed.
3. Collapsing identities x = y and x = e are closed.
4. The identity x = y · yy is collapsing but not closed.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Basic terms
A quasigroup term b is basic if:
b is a variable (b is linear); or
b = x ◦ x for some variable xand some ◦ ∈ {·, /, \} (b is quadratic).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Representation
Theorem (Fempl–Madjarevic, Krapez)
Let s = t be nontrivial closed quasigroup identity. Then there arelinear terms L1[x1, . . . , xk ], L2[xk+1, . . . , xn] such thats = L1[b1, . . . , bk ]t = L2[bk+1, . . . , bn]where bi (1 ≤ i ≤ n) are basic terms and
s = t ⇔∧n
i=1(bi = e).
If at least one of bi is linear, then s = t is collapsing.Otherwise, s = t is equivalent to the one of:(eQ), (Qe), (U), (1), (eU), (Ue), (U1).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Translations
Definition
For a given quasigroup (Q; ·, /, \), translations are:
La(x) = a · x L−1a (x) = a\x
Ra(x) = x · a R−1a (x) = x/a
Pa(x) = x\a P−1a (x) = a/x
It is convenient to have numerical values for translations:
ε L R P P−1 R−1 L−1
value 0 1 2 3 4 5 6
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Definition
We shorten:D(x , y) = γ−1A(αx , βy)
to:D = A(α, β, γ).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Belousov’s derivatives
Aa = A(La, ε, La) (Right derivative)
aA = A(ε,Ra,Ra) (Left derivative)
Aa
= A(Pa,Pa, ε) (Middle derivative)
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Derivatives
D = A(α, β, γ) is derivative if:
α, β, γ are bijections
one of them is ε.
Da = A(α, β, γ) is inner derivative (relative to a ∈ Q) if:
the one of α, β, γ is ε
the other two are translations (by a).
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Problem
ProblemDa = A implies the existence of an autotopy of A.Does it imply the existence of some unit?
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Numbers for identities
For every identity Da = A we have a three digit number pqr(where one of p, q, r is 0 (which stands for ε) and the other twoare numbers between 1 and 6 (and they stand for appropriatetranslations by a). If one of p, q, r is n it means that it stands forall numbers from 1 to 6.
For example, the identity Aa = A is represented by the number 101.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Results
Theorem (Krapez, Shcherbacov)
pqr ⇒ (eQ) forpqr ∈ {n01, n06, n10, n60, 102, 105, 130, 140, 602, 605, 630, 640}.
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups
DedicationIntroduction
Problems1st family – Identities similar to weak associativity
2nd family – closed identities3rd family – derivative autotopies
Results
Analogously:
pqr ⇒ (Qe) forpqr ∈ {0n2, 0n5, 2n0, 5n0, 021, 026, 051, 056, 320, 350, 420, 450}.
pqr ⇒ (U) forpqr ∈ {03n, 04n, 30n, 40n, 013, 014, 063, 064, 203, 204, 503, 504}.
pqr ⇒ (1) for pqr ∈ {210, 260, 510, 560}.
pqr ⇒ (eU) for pqr ∈ {301, 306, 401, 406}.
pqr ⇒ (Ue) for pqr ∈ {032, 035, 042, 045}.
pqr ⇒ (U1) for no pqr .
Aleksandar Krapez (with J. Fempl–Madjarevic and V. A. Shcherbacov)Units in quasigroups