GROUPOIDS WITH SEMIGROUP OPERATORS AND ADDITIVE ENDOMORPHISM A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS POR THE DEGREE OF MASTER OF SCIENCE BY FRED HENDRIX HUGHES DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA JUNE 1965 TABLE OF CONTENTS Chapter Page I' INTRODUCTION. 1 II GROUPOIDS WITH SEMIGROUP OPERATORS 6 - Ill GROUPOIDS WITH ADDITIVE ENDOMORPHISM 12 BIBLIOGRAPHY 17 ii 4 CHAPTER I INTRODUCTION A set is an undefined termj however, one can say a set is a collec¬ tion of objects according to our sight and perception. Several authors use this definition, a set is a collection of definite distinct objects called elements. This concept is the foundation for mathematics. How¬ ever, it was not until the latter part of the nineteenth century when the concept was formally introduced. From this concept of a set, mathematicians, by placing restrictions on a set, have developed the algebraic structures which we employ. The structures are closely related as the diagram below illustrates. Quasigroup Set The first structure is a groupoid which the writer will discuss the following properties: subgroupiod, antigroupoid, expansive set homor- phism of groupoids in semigroups, groupoid with semigroupoid operators and groupoids with additive endormorphism. Definition 1.1. — A set of elements G - f x,y,z } which is defined by a single-valued binary operation such that x o y ■ z é G (The only restriction is closure) is called a groupiod. Definition 1.2. — The binary operation will be a mapping of the set into itself (AA a direct product.) 1 2 Definition 1.3» — A non-void subset of a groupoid G is called a subgroupoid if and only if AA C A. It is called an antigroupoid if and only if AA C. G-A (the complement of A. ) All other non-void sub¬ sets are said to be expansive. a a a c c b a b b a Subsets c c b c b d d d d d If G « £ a,b,c,d | then £a>c"f and £a,b| are subgroupoids and is an expansive set and is the indempotent groupoid. Definition l.U. — An element e of a groupoid is an indempotent if and only if e^ = e Theorem 1.1, — A groupoid G contains no antigroupoid if and only if every element of G is an indempotent. Proof: The proof follows directly from definitions 1.3 and l.lu Since G is idempotent, this implies that some products are contained in G. Since this true AA G-A. Hence, by definition it is not an antigroupoid. Definition 1.5. — A groupoid G is cyclic if and only if there exist an a in G such that G is the collection of sill powers of a. Theorem 1.2. — A groupoid containing no proper subgroupoid is cylic. Proof: Let A 6 G and A ^ G G contains at least one element different from A. G is cyclic, therefore it contains elements • a. n , , , / « I a. B-« e = x, x,I x, .... x Let A » jx, x, x, .... % 2r Now consider the mapping AA into A, in particular a?*" x ®= x 1’ ^ A. Hence A is not a subgroupiod of G for AA A. Therefore, G is cyclic and contains no proper subgroupoid. Theorem l.lj, — Every non-void subset of a groupoid G is a sub¬ groupoid or an antigrohpoid if and only if each non-void subset of cardi¬ nality less than or equal to five is an antigroupoid. Proofs Let A be an expansive subset of G. Then -j elements a, aA a, a^ of A such that a, a^ is in A and a, a¥ are in G-A. Then B = is an expansive set of cardinality less than or equal to five. This condition follows easily from Definition 1.3. Theorem 1.5. — Every non-empty subset of a groupoid G is a subgroupiod if and only if xy = x,ar y for all x, y in G. Proofs By Definition 1.3 , this condition must exist for any sub¬ groupoid. Suppose A any subset of G with a mapping o{ . If AA ^ G, then A is either an antigroupoid or an expansive set. But AA (Z- A, there¬ fore a subgroupoid. Theorem 1.6. — Every proper subset of a groupoid G is an antigroupoid if and only if G has exactly two elements x, y such that x = y or y = x or G is the one element groupoid. Proofs Suppose G has two distinct elements x,y. Then since y is in Similarly, x** * y and if q is in (G - -[x} ) Then q** = x = y. Theorem 1,7» — If such that K (Z K is the collection of expansive sets then the U A* is a subgroupoid or an expansive set. If is a collection of antigroupoid, then the f \ AK is an enpty h or an antigroupoid. I Proof of the . Suppose x é L-/# —7 x is a member of at least one . The Q AKfK^ K. Now since x does not be¬ long to all AK "J no mapping xx = x or xy = x. Hence the union is an expansive set. Proof of Part 2 s To show that the n AK is not empty, let X €^ A^ -=7 x é AK V AK sf\A =, (f) . Suppose that x HQ. A^ x does not belong to all AK , therefore } no mapping AA C A. Hence an antigroupoid. Theorem 1.8. — A semigroup (associative groupoid 0 which is the union of an antigroupoid is infinite.) Proofs Every finite semigroup had in idempetent. Now some exampless Example 1.1. — Let G be the intergers under subtraction a o b = a-b the odd integers are antigroupoid and the even integers are subgroupoid and the odd integers are expansive. If G is the integer under ordinary multiplication, the negative integers form an antigroupoid. Example 1.2. — Let G be the positive integers under the multiplication ab » 1 if a # b aa « a + 1 G is a cyclic groupoid with generator 1 and has no proper subgroupoid since every subgroupoid contains 1. G is abelian and (11) 1 1(11). Clearly, G is nonassociative. Definition 1.6. — An element a of a groupoid is generally prime if and only if jf* no b and c in G ^ a » be. Theorem 1.7« -- A groupoid G which contains a generally prime ele¬ ment has a non-trivial abelian semigroup as its homomorphic image. Proofs Clearly GG £ G. Let P ** G-GG, the set of generally prime 5 elements of G. We note that PG C. G-P and GP CL G-P and that G-P is a subgroupoid. Let w be any element not in G, and let G* = P(J », De¬ fine xy = w for all x, y in G. Then G is a non-trivial abelian semigroup if we define f (x) - X for in P f (x) = w for x in GP then f is a homomorphism of G onto G. Example 1.3» — Let G be the interval numbers under 2 the multiplication a a b » ab for all a,b £ G. The set P of generally * Hence, G with x* y = w for all x, y in G * . Thus, G is a nonassociative and non-abelian groupoid which has a non-trival abelian semigroup as its homomorphic image. Definition 1.7. ~ A necessary and sufficient condition for a groupoid G have the two element group as its homomorphic image is that G contain a partial ideal I and that G-l be a subgroupoid. Proof î Let the 2-group G consist of 0 and 1 with the operation + i.e., 0+0=0, l+0=0+l=l, and 1+1=0. Define F (x) = 1 for all for x £ X and f (x) = 0 for x in G-I. It is easily shown that is a p homomorphism of G onto G . Conversely, suppose that f is a homomorphism of G onto G* . Then I = f”1 (1) is a partial ideal of G- and g 1-1 (0) is a subgroupoid. Theorem 1.8. — If I is a partial ideal of a semigroup S, S-I is a semigroup. If S is a group, S - I is a group of index two in S. Proof: Suppose a,b are in S-I and ab is in I. If c is in I, (ab)c is in S-I and a(bc) is in I contradicting associativity. If s is a group with identity e, e is in S-I and hence a in S-I d? a"1 is in S-I and x(S-I) = I for all x in I. CHAPTER II . GROUPOIDS WITH SEMIGROUP OPERATORS Review of Terms. — Isomorphism, if there exist a one to one cor¬ respondence between two sets such that operations are preserved, then we see that the sets are isomorphic. Automorphism, if there exist an isomor¬ phism on a set such that every element of the set goes over to itself, then the mapping is called an automorphism. Homomorphism, if there exist a many one mapping, correspondence between two sets such that the opera¬ tion is preserved or there exist a single value transformation such that it maps a set onto a set, then we say the sets are homomorphic. Endomor¬ phism, if a set is homomorphic to another set where the set is contained in the set, then the set is to be endomorphic. Semigroup Operators. ~ Let G be a groupoid and suppose that p is a commutative semigroup.