幂等弱生成的正则半群

Lu'is Oliveira
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引用次数: 2

摘要

如果集合X不存在包含X的正则子群,则正则半群是由集合X弱生成的。本文研究了幂等函数弱生成的正则半群。证明了存在一个由|X|幂等幂弱生成的正则半群FI(X),使得其他所有由|X|幂等幂弱生成的正则半群都是FI(X)的同态像。给出了半群FI(X)的定义$\langle G(X),\rho_e\cup\rho_s\rangle$,并研究了它的结构。虽然对于$|X|\geq 2$,每个集合$G(X)$、$\rho_e$和$\rho_s$都是无限的,但我们证明了字问题是可判定的,因为每个同余类都有一个规范形式。如果$FI_n$表示$|X|=n$的FI(X),我们也证明$FI_2$包含所有$FI_n$的副本作为子半群。因此,我们得到(i)由有限幂等生成的所有正则半群(包括所有有限幂等生成的正则半群)弱可除$FI_2$;(ii)所有有限半群可除$FI_2$。
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Regular semigroups weakly generated by idempotents
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\langle G(X),\rho_e\cup\rho_s\rangle$ and its structure is studied. Although each of the sets $G(X)$, $\rho_e$, and $\rho_s$ is infinite for $|X|\geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.
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