Let d!= {P, , . . . . P,} be a family of sets. A partial order P(@, <) on CD is naturally defined by the condition P, < 5 iff P, is contained in 4. When the elements of Cg are disks (i.e. circles together with their interiors), P(@, <) is called a circle order; if the elements of Cg are n-polygons, P(cD, <) is called an n-gon order. In this paper we study circle orders and n-gon orders.The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n B 4 partial orders of dimension n which are not circle orders. Also for every n> 3, we prove that there are partial orders of dimension 2n +2 which are not n-gon orders.Finally, we prove that every partial order of dimension 62n is an n-gon order.

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A Banach algebra ; is AMNM if whenever a linear functional φ on ; and a positive number δ satisfy Qφ(ab)kφ(a) φ(b)Q δRaR:RbR for all a, b ? ;, there is a multiplicative linear functional ψ on ; such that RφkψR l o(1) as δ 0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM.If ; is a Banach algebra, ψ is a (possibly zero) multiplicative linear functional on ;, and τ is a bounded linear functional on ; of norm RτR l σ, a trivial calculation shows that the linear functional φ B ψjτ is δ-multiplicati e with δ l (3jσ) σ in the sense thatLoosely, if the linear functional φ on ; is near a multiplicative linear functional, then it is approximately multiplicati e. The notion of approximately multiplicative linear functional, or even operator, is discussed in Krzysztof Jarosz' monograph [1], in which it is shown (Proposition 5.5) that if the linear functional φ on ; is δmultiplicative, then RφR 1jδ, which is a step in the direction of proving that φ must be near a multiplicative linear functional. Jarosz poses the problem (Problem 5, page 111) : if ; is a Banach algebra (respectively, uniform algebra) and φ is a δmultiplicative linear functional on ;, must there exist a multiplicative linear functional on ; such that RφkψR l o(1) as δ 0 ? In Barry Johnson's definitive study [3] of approximately multiplicative linear functionals, he calls commutative Banach algebras for which the answer to this question is affirmative AMNM (approximately multiplicative is near multiplicative) algebras. Johnson shows that many classical commutative Banach algebras are AMNM, while presenting (Example 9.1) a commutative semisimple Banach algebra that is not AMNM. In particular, he shows (Theorem 7.1 et seq.) that polydisc algebras are AMNM, but leaves open the question of whether H _ , or indeed every uniform algebra, is AMNM. Our purpose here is to fill part of this gap by producing uniform algebras that are not AMNM. Very recently, Jarosz [2] has filled other portions of this gap, proving, for instance, that ball algebras and singly generated uniform algebras are AMNM (in particular, his proof that the disc algebra is AMNM is far simpler than Johnson's). Jarosz' paper, which contains both positive and negative results, includes some very beautiful work (for instance, the ball algebras proof ).Our main result in this paper is the following.