Question

Question 9. Let (A-) be a binary structure. When the book defined identity, it meant 2 sided identity, but it is also possible to talk about one sided (right and left) identities. Come up with a reasonable definition of the terms: left identity (denoted by ez) and right identity denoted by eR). (a) Is it true that, if a left (respectively, right) identity exists, then it is unique? If it is true, prove it; if it is false, provide a counterexample, and also explain where the proof of Theorem 3.13 falls apart (b) Can a binary structure have different left and right identities? In other words, suppose that et and eR exist, is it possible that e eR? Give an example of such a binary structure on prove that it is impossible. No credit will be given for this question, if incorrect definitions are given at the start, so if in doubt, talk to a classmate, your instructor, a TA etc. For part (b), start by rereading the proof of Theorem 3.13.]

Answer 1

If the left identity exist then it is may not be unique ex: - aob = a^b forall a, b elementof N then eR elementof N exist aoe_L = a Rightarrow a^eL = a^� Rightarrow e_B = 1 rightarrow unique i.e forall a elementof N exist 1 elementof N exist ao1 = a, e_L not exist But when we take a nonempty set �s� with at least two elements the binary operation on �5� defined as aob = b, then every element of s is a left identity of (s, o) and aob = a then every element of �5� is a right identity of �s� Therefore the left (right) Identity exist it may not be unique (s, 0) is a binary structure and e_L, e_R be the left and right identifies in �s� then e_L = E_R (if both exists) proof: - e_L, e_R elementof S and e_L is left identity Rightarrow e_Loe_R = e_R - (a)