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.