### $3$-configurations with simple edge basis and their corresponding quasigroup identities

There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.

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There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.

Using the Kramer-Mesner method, $4$-$(26,6,\lambda )$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\{30,51,60,81,90,111\}$ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda )$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\{51,60,81,90,111\}$. With $\lambda $ in the set $\{60,81\}$, there exist designs which possess all three considered groups...

In the present paper we construct the accompanying identity $\widehat{I}$ of a given quasigroup identity $I$. After that we deduce the main result: $I$ is isotopically invariant (i.e., for every guasigroup $Q$ it holds that if $I$ is satisfied in $Q$ then $I$ is satisfied in every quasigroup isotopic to $Q$) if and only if it is equivalent to $\widehat{I}$ (i.e., for every quasigroup $Q$ it holds that in $Q$ either $I,\widehat{I}$ are both satisfied or both not).

After describing a (general and special) coordinatization of $k$-nets there are found algebraic equivalents for the validity of certain quadrangle configuration conditions in $k$-nets with small degree $k$.

Our short note gives the affirmative answer to one of Fishburn’s questions.

Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.

For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ωk, the set of all k-subsets of Ω = 1, 2, …, 2k +1, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k)...