SMARANDACHE ALGEBRAIC STRUCTURES
by Raul Padilla
A few notions are introduced in algebra in order to better study the
congruences. Especially the Smarandache semigroups are very important
for the study of congruences.
1) The SMARANDACHE SEMIGROUP is defined to be a semigroup A such that a
proper subset of A is a group (with respect to the same induced
operation).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
For example, if we consider the commutative multiplicative group
SG = {18^2, 18^3, 18^4, 18^5} (mod 60)
we get the table:
x | 24 12 36 48
--- |-------------
24 | 36 48 24 12
12 | 48 24 12 36
36 | 24 12 36 48
48 | 12 36 48 24
Unitary element is 36.
Using the Smarandache's algorithm we get that
18^2 is congruent to 18^6 (mod 60).
Now we consider the commutative multiplicative semigroup
SS = {18^1, 18^2, 18^3, 18^4, 18^5} (mod 60)
we get the table:
x | 18 | 24 12 36 48
----|----|------------
18 | 24 | 12 36 48 24
----|----|------------
24 | 12 | 36 48 24 12
12 | 36 | 48 24 12 36
36 | 48 | 24 12 36 48
48 | 24 | 12 36 48 24
Because SS contains a proper subset SG, which is a group, then SS is a
Smarandache Semigroup. This is generated by the element 18. The
powers
of 18 form a cyclic sequence: 18, 24,12,36,48, 24,12,36,48,... .
Similarly are defined:
2) The SMARANDACHE MONOID is defined to be a monoid A such that a proper
subset of A is a group (with respect with the same induced operation).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
3) The SMARANDACHE RING is defined to be a ring A such that a proper
subset of A is a field (with respect with the same induced operation).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
We consider the commutative additive group M={0,18^2,18^3,18^4,18^5}
(mod 60) [using the module 60 residuals of the previous powers of 18],
M={0,12,24,36,48}, unitary additive unit is 0.
(M,+,x) is a field.
While (SR,+,x)={0,6,12,18,24,30,36,42,48,54} (mod 60) is a ring whose
proper subset {0,12,24,36,48} (mod 60) is a field.
Therefore (SR,+,x) (mod 60) is a Smarandache Ring.
This feels very nice.
4) The SMARANDACHE SUBRING is defined to be a Smarandache Ring B which
is a proper subset of s Smarandache Ring A (with respect with the same
induced operation).
5) The SMARANDACHE IDEAL is defined to be a Smarandache subring
that absorbs (to the left, to the right, or both) the whole ring
(with respect with the same induced operation).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
6) The SMARANDACHE LATTICE is defined to be a lattice A such that
a proper subset of A is a Boolean algebra (with respect with the same
induced operations).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
7) The SMARANDACHE FIELD is defined to be a field (A,+,x) such that a
proper subset of A is a K-algebra (with respect with the same induced
operations, and an external operation).
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
8) The SMARANDACHE R-MODULE is defined to be an R-MODULE (A,+,x) such
that a proper subset of A is a S-algebra (with respect with the same
induced operations, and another "x" operation internal on A), where R is
a commutative unitary Smarandache ring and S its proper subset field.
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
9) The SMARANDACHE K-VECTORIAL SPACE is defined to be a K-vectorial
space (A,+,.) such that a proper subset of A is a K-algebra (with
respect with the same induced operations, and another "x" operation
internal on A), where K is a commutative field.
By proper subset we understand a set included in A, different from the
empty set, from the unit element -- if any, and from A.
References:
[1] Castillo, J., "The Smarandache Semigroup", , II Meeting of the project
'Algebra, Geometria e Combinatoria', Faculdade de Ciencias da
Universidade do Porto, Portugal, 9-11 July 1998.
[2] Padilla, Raul, "Smarandache Algebraic Structures", , Delhi, India, Vol. 17E, No. 1, 119-121, 1998.
[3] Padilla, Raul, "Smarandache Algebraic Structures", , USA, Vol. 9, No. 1-2, 36-38, Summer 1998.
[4] Padilla, Raul, "On Smarandache Algebraic Structures", American
Research Press, to appear.
Presented to the , Universidade do
Minho, Braga, Portugal, 18-23 June, 1999.