From affine to projective geometry via convexity.

*(English)*Zbl 0577.08003
Universal algebra and lattice theory, Proc. Conf., Charleston/S.C. 1984, Lect. Notes Math. 1149, 255-269 (1985).

[For the entire collection see Zbl 0563.00005.]

The main result of the paper is the Theorem 2.4, as follows: Let K be a field, and let (E,K,P) be an affine space over K. Then the projective geometry (L(E),’) is the largest \(\Omega_ K\)-semilattice quotient of the algebra \(((E,K,P)S,\Omega_ K)\) of affine subspaces of (E,K,P).

Three sections give the proof of the various cases of the theorem above: Section 3 deals with K of characteristic 0, Section 4 with K of odd characteristic, and Section 5 with K of characteristic 2. In section 5 general results of Płonka are applied. A method of determining a basis for the identities of the variety determined by all its regular identities is presented. The property that the regularization of a variety is finitely based if the variety is finitely based is observed (under the assumption that there is a strongly nonregular identity satisfied in the given variety and the considered type is finite).

The main result of the paper is the Theorem 2.4, as follows: Let K be a field, and let (E,K,P) be an affine space over K. Then the projective geometry (L(E),’) is the largest \(\Omega_ K\)-semilattice quotient of the algebra \(((E,K,P)S,\Omega_ K)\) of affine subspaces of (E,K,P).

Three sections give the proof of the various cases of the theorem above: Section 3 deals with K of characteristic 0, Section 4 with K of odd characteristic, and Section 5 with K of characteristic 2. In section 5 general results of Płonka are applied. A method of determining a basis for the identities of the variety determined by all its regular identities is presented. The property that the regularization of a variety is finitely based if the variety is finitely based is observed (under the assumption that there is a strongly nonregular identity satisfied in the given variety and the considered type is finite).

Reviewer: Ch.Herrmann

##### MSC:

08A30 | Subalgebras, congruence relations |

51A05 | General theory of linear incidence geometry and projective geometries |

08B05 | Equational logic, Mal’tsev conditions |