SMARANDACHE ALGEBRAIC STRUCTURES
by Raul Padilla
A few notions were introduced in algebra [5] 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. [5] Smarandache, Florentin, Special Algebraic Structure (1973), Arizona State University, Special Collections, Tempe, AZ, USA.
Presented to the , Universidade do
Minho, Braga, Portugal, 18-23 June, 1999.