Abstract
We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a “root” plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz–Barlotti types for these planes are then linked to the Lenz–Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz–Barlotti types II.1 and V.1.
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Robert S. Coulter was partially supported by NSF Grant Award #1106938.
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Coulter, R.S., Matthews, R.W. Closure planes. J Algebr Comb 43, 735–749 (2016). https://doi.org/10.1007/s10801-015-0655-5
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DOI: https://doi.org/10.1007/s10801-015-0655-5