Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s : M × M −→ M × M such that s 23 s 13 s 12 = s 12 s 23 , where s 12 = s × id M , s 23 = id M ×s and s 13 = (id M ×τ )s 12 (id M ×τ ), and τ is the flip map, i.e., the permutation on M × M given by τ (x, y) = (y, x), for all x, y ∈ M .In this paper we give a complete description of the set-theoretical solutions of the form s(x, y) = (x · y, x * y) when either (M, ·) or (M, * ) is a group; moreover, we raise some questions.
In this paper, we give a characterization of the quasi-linear left cycle sets [Formula: see text] with [Formula: see text] via unitary metahomomorphisms and a complete description of those with [Formula: see text] improving the results obtained in [F. Catino and M. M. Miccoli, Construction of quasi-linear left cycle sets, J. Algebra Appl. 14(1) (2015), Article ID:1550001, 1–7]. Moreover, we develop a theory of dynamical extensions of quasi-linear left cycle sets to provide new set-theoretic solutions of the Yang–Baxter equation that are non-degenerate, involutive and multipermutational.
We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations $$+$$ + and $$\circ $$ ∘ such that both $$(S,+)$$ ( S , + ) and $$(S, \circ )$$ ( S , ∘ ) are inverse semigroups and $$\begin{aligned} a \circ \left( b+c\right) = \left( a\circ b\right) - a + \left( a\circ c\right) \qquad \text {and} \qquad a\circ a^- = - a + a \end{aligned}$$ a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all $$a,b,c \in S$$ a , b , c ∈ S , where $$-a$$ - a and $$a^-$$ a - are the inverses of a with respect to $$+$$ + and $$\circ $$ ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on $$S\times S$$ S × S . In addition, we provide some methods to construct weak braces.
In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad (A) ⊆ Fix (A). Moreover, among these cycle sets, we give a complete description of those for which Fix (A) = Soc (A) and the underlying additive group is cyclic. Using such cycle sets, we obtain left non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation which are different from those obtained in [P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999) 169–209; P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709–719].
We prove that the upper central chain of the multiplicative group of a local ring R coincides with the chain of the multiplicative group of terms of the upper central chain of the associated Lie ring of R.
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