Simple power associative loops with exactly one covering.
A loop has a `covering' by subloops if it is the set-theoretic union of proper subloops. A covering of a loop \((L,\cdot)\) is `irredundant' if each subcollection of those loops fails to cover \(L\). A loop is `uniquely covered' if it has exactly one irredundant covering. We note that for a loop having `exactly one covering' is a stronger condition than being uniquely covered. In this paper the author presents a class of power associative loops uniquely covered or in some more specific assumptions having exactly one cover. The construction of this family of loops proceeds as follows: let \(S\) be a non empty set endowed with two binary operations ``+'' and ``\(\cdot\)'' such that \((S,+)\) is a loop with identity 0 and \((S^*:=S\setminus\{0\},\cdot)\) is a quasigroup and let \((Q,\odot)\) be an idempotent quasigroup. Then the set \({\mathcal L}^{(Q)}(S):=S^*\times Q\cup\{{\mathbf 1}\}\) (where \({\mathbf 1}\notin S^*\times Q\)) can be furnished with a new binary operation of loop with identity \({\mathbf 1}\). When \((S,+)\) is a group then \({\mathcal L}^{(Q)}(S)\) is power associative. The author investigates such loops when \((S,+,\cdot)\) is a field with \(|S|>2\) (the case \(S=\text{GF}(2)\) and \(|Q|=3\) gives that \({\mathcal L}^{(Q)}(S)\) is the Klein 4-group). Assuming that \((S,+,\cdot)\) is a finite field with \(|S|>2\) and that \((Q,\odot)\) is a two-quasigroup (i.e. a non trivial two-generated quasigroup having a doubly transitive automorphism group) the author proves that \({\mathcal L}^{(Q)}(S)\) is uniquely covered. If moreover the field \((S,+,\cdot)\) is of prime order then \({\mathcal L}^{(Q)}(S)\) has exactly one covering by proper subloops. Adding more specific properties to the finite field \(S\) or to the two-quasigroup \(Q\), the loop \({\mathcal L}^{(Q)}(S)\) becomes simple or satisfies the so-called `strong Lagrange property'. In particular, if \(S\) is a field of prime order and \(Q\) is a finite round-robin two-quasigroup and \(|Q|>3\) or \(|S|>2\), then \({\mathcal L}^{(Q)}(S)\) is an Abelian simple loop with exactly one covering by proper subloops.
scientific article; zbMATH DE number 2119243