AXIOMATIC DUALITY BETWEEN ASSOCIATIVITY AND INVERTIBILITY JODMOS HORON SHA256 checksum of the author’s real name: DC8D57E03718CEEF AF929009C0AED7F7 75E6C5F155B813F3 F12534C770D385F0 Definition 1. We consider the two following axioms concerning a ternary relation , laid out below as inference rules in a custom notation: xy yz xv uz xv uz xy yz u v w w w w u v Remark 2. These inference rules are supposed to be understood in the following way: all variables that appear in the premises (what’s above the inference line) are considered universally quantified; all variables that appear in the conclusions (below the inference line) but not in the premises are considered existentially quantified, the scope of that latter quantification being the conclusions. Which means that these axioms can respectively be rewritten, in more usual logical notations, as follows: ∀x, y, u, z, v, xy u ∧ yz ∀x, u, z, v, w, xv w ∧ uz v =⇒ w =⇒ ∃w, xv ∃y, xy w ∧ uz w u ∧ yz v Definition 3. We will say that the ternary relation interprets a binary composition law ◦ when ab c will be synonymous to a ◦ b = c, universally in a, b and c. A binary operation will be said to admit a unit when there exists e such that x ◦ e = x and e ◦ x = x, universally in x. Theorem 4. Under the additional assumption that the relation interprets a binary law admitting a unit, these axioms axiomatise groups. Remark 5. Given that the two axioms can be obtained by reversing the premises and conclusions in the notation we have developed, this theorem showcases a duality between associativity and inversion in a sense yet to be made more precise. Proof. We will first check that a structure S endowed with such a ternary relation satisfying the two axioms and interpreting a binary operation admitting a unit is a group. And then check the converse of that statement. Date: 15th March 2021. 1 AXIOMATIC DUALITY BETWEEN ASSOCIATIVITY AND INVERTIBILITY 2 Recall that the conventional axiomatisation of groups is that of a binary law that is associative, admits a unit, and inverses. What remains to be checked therefore is associativity and invertibility. Let therefore x, y and z be three items of our structure S. As it is assumed that interprets a binary law ◦, we may define u to be u := x ◦ y and v to be v := y ◦ z. The first axiom can therefore be rewritten as follows: x◦y =u y◦z =v x◦v =w u◦z =w The premises being valid by construction, this proves the existence of a w such that x ◦ v = w and such that u ◦ z = w. Which proves associativity. Let now γ be any item of our structure X . We have assumed the existence of a unit e. We may therefore specialise the second axiom with the following assignments: x → γ, z → γ, u → e, v → e and w → γ. Which yields: γ◦e=γ e◦γ =γ γ◦y =e y◦γ =e As e is known to be a unit, the premises hold. Which implies the existence of a y such that the conclusions holds. Which shows that y is an inverse to γ and therefore that all items of X admit inverses. Hence S is a group, and any structure satisfying these two axioms under the assumption that interprets a binary operation admitting a unit is a group. Conversely, suppose that G is a group with a neutral element e and a composition law ◦. We then interpret to be a ternary relation where ab c means a ◦ b = c. We want to show that the two axioms are verified as that would prove our theorem. We now check the first axiom, and therefore suppose that xy u and yz v hold. Which means that x ◦ y = u and y ◦ z = v. We then define w0 and w1 as w0 := x · v and w1 := u · z. As G is a assumed to be a group, the composition law is associative, and we then know that w0 = w1 , and we call w this item. We then have xv w and uz w. Which proves that the first axiom holds. We now check the second axiom, and therefore suppose that xv w and uz w. Which means that x ◦ v = w = u ◦ z. As G is assumed to be a group, it admits inverses, and we then have x ◦ v ◦ z −1 = u and v = x−1 ◦ u ◦ z. Defining y0 and y1 as y0 := v ◦ z −1 and y1 := x−1 ◦ u, we then get x ◦ y0 = u and y1 ◦ z = v. In other words, xy0 u and y1 z v hold. But we also know that the following also holds: y0 = v ◦ z −1 = x−1 ◦ x ◦ v ◦ z −1 = x−1 ◦ u ◦ z ◦ z −1 = x−1 ◦ u = y1 We may therefore define y to be that element y0 = y1 , and we then know that u and yz v hold. Therefore the second axiom holds. The converse is therefore proven: all groups satisfy the two axioms. This shows that groups are therefore axiomatised by these two axioms under the assumption that they interpret a composition law admitting a unit. Which proves the theorem. xy Email address: [email protected]
