A note on maximal subgroups of free idempotent generated semigroups over bands
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A NOTE ON MAXIMAL SUBGROUPS OF FREE IDEMPOTENT GENERATED SEMIGROUPS OVER BANDS Igor Dolinka1 Department of Mathematics and Informatics, University of Novi Sad Trg Dositeja Obradovi´ca 4, 21101 Novi Sad, Serbia E-mail: [email protected] (Received October 20, 2010; Accepted July 13, 2011) [Communicated by M´ aria B. Szendrei]
Abstract We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.
Let S be a semigroup, and let E = E(S) be the set of its idempotents; in fact, E, along with the multiplication inherited from S, is a partial algebra. It turns out to be fruitful to restrict further the domain of the partial multiplication defined on E by considering only the pairs e, f ∈ E for which either ef ∈ {e, f } or f e ∈ {e, f } (i.e. {ef, f e} ∩ {e, f } 6= ∅). Note that if ef ∈ {e, f } then f e is an idempotent, and the same is true if we interchange the roles of e and f . Such unordered pairs {e, f } are called basic pairs and their products ef and f e are basic products. The free idempotent generated semigroup over E is defined by the following presentation: IG(E) = hE | e · f = ef such that {e, f } is a basic pair i. Here ef denotes the product of e and f in S (which is again an idempotent of S), while · stands for the concatenation operation in the free semigroup E + (also to be interpreted as the multiplication in its quotient IG(E)). An important feature of IG(E) is that there is a natural homomorphism from IG(E) onto the subsemigroup of S generated by E, and the restriction of φ to the set of idempotents of IG(E) is a basic-product-preserving bijection onto E, see, e.g., [5], [9], [13]. Mathematics subject classification numbers: 20M05, 20M10, 20F05. Key words and phrases: free idempotent generated semigroup, band, maximal subgroup. 1
The support of the Ministry of Education and Science of the Republic of Serbia, through Grant No. 174019, is gratefully acknowledged. 0031-5303/2012/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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I. DOLINKA
An important background to these definitions is the notion of the biordered set [7] of idempotents of a semigroup and its abstract counterpart. The biordered set of idempotents of S is just a partial algebra on E(S) obtained by restricting the multiplication from S to basic pairs of idempotents. In this way we have that if B is a band (an idempotent semigroup), then, even though there is an everywhere defined multiplication on E(B) = B, its biordered set [3] is in general still a partial algebra. Another way of treating biordered sets is to consider them as relational structures (E(S), ≤(l) , ≤(r) ), where the set of idempotents E(S) is equipped by two quasi-order relations defined by e ≤(l) f if and only if ef = e, e ≤(r) f if and only if f e = e. One of the main achievements of [4], [5], [9] is the result that the class of biordered sets considered as rela
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