Triangular Scheme Revisited in the Light of n-permutable Categories

2021

EJMS

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The purpose of this paper is two-fold. A first and more concrete aim is to give new characterizations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the little Pappian Theorem involving reflexive and positive relations. A second and more abstract aim is to show that every finitely complete category E satisfying the n-scheme is locally anticommutative.

Section: Let Us Begin With the Following Observationmentioning

confidence: 94%

Anticommutativity and n-schemes

2021

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“…Following [1], given any pair (R, S) of reflexive relations on an object X in a regular category E, let us denote by (R; S) n ; n ≥ 2 the alternate composition RSRS • • • of length n which is a reflexive relation as well. Clearly we have: (R, S) n ≤ (R, S) n+1 and (S, R) n ≤ (R, S) n+1 .…”

Section: Equivalence Relations In N-permutable Categoriesmentioning

confidence: 99%

Geometrical Methods in Goursat Categories

2021

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The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, hold in any regular Mal'tsev categories. We prove that Mal'tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, that is classically expressed in terms of four congruences R, S1, S2 and T, and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3. This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2n+1-permutable category $\mathcal{E}$, the category Equiv$(\mathcal{E})$ of equivalence relations in $\mathcal{E}$ is also a 2n+1-permutable category.

“…Consider G = Z 2 × Z 6 and let U 1 = 1, U 2 = {(0, 0), (1, 0), (0, 2), (1, 2), (0, 4), (1, 4)}, U 3 = {(0, 0), (1, 0), (0, 3), (1, 3)}, U 4 = {(0, 0), (1, 0)}, U 5 = {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5)}, U 6 = {(0, 0), (0, 2), (0, 4)}, U 7 = {(0, 0), (0, 3)}, U 8 = {(0, 0), (0, 2), (0, 4),(1,1),(1,3),(1,5)}, U 9 = {(0, 0), (1, 3)}, U 10 = Z 2 × Z 6 . See Figure for the subgroup lattice of G.…”

mentioning

confidence: 99%

Decomposition of Goursat Matrices and Subgroups of Zm x Zn

2021

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Given the number of subgroups of Zm x Zn, we deduce the Goursat matrix. The purpose of this paper is two-fold. A first and more concrete aim is to demonstrate that the triangular decomposition of the Goursat matrix may also be written out explicitly, and furthermore that the same is true of the inverse of these triangular factors. A second and more abstract aim provides a containment relation property between subgroups of a direct product . Namely, if U2 ≤ U1 ≤ Zm x Zn, we provide necessary and sufficient conditions for U2 ≤ U1.