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Principal-ideal bands. (English) Zbl 0938.20044

Let \(S\) be a band. \(S\) is called a principal-ideal band if \(S\) is with identity in which every ideal is principal. In the case of matrix bands (idempotent subsemigroups of the multiplicative semigroup of \(M_n(\mathbb{F})\)), it has been shown that such bands are triangularizable [H. Radjavi, J. Oper. Theory 13, 63-71 (1985; Zbl 0581.47026)]. Let \(\mathcal S\) be a principal-ideal band in \(M_n(\mathbb{F})\), \(\{E_i\}_{i=0}^r\) a maximal family of commuting idempotents in \(\mathcal S\). Without loss of generality the authors assume that \(E_0=0\), \(E_r=I\), \(\text{rank }E_i<\text{rank }E_{i+1}\), and consider the decomposition \(\mathbb{F}^n=M_1+M_2+\cdots+M_r\). Each element of the band can be represented as an \(r\times r\) block matrix with respect to the above decomposition. Let \({\mathcal S}(m)=\{S\in{\mathcal S}:\Delta(S)=[E_m]\}\) (the equivalence class containing \(E_m\)) and \({\mathcal S}_{pq}(m)=\{S_{pq}:S\in S(m)\}\) for \(m,p,q=1,2,\ldots,r\), where \(S_{pq}\) denotes the \((p,q)\)-th entry of the \(r\times r\) block matrix representation \(S=(S_{pq})\) of \(S\) with respect to the above decomposition.
In this paper, the authors prove that an element of \({\mathcal S}(m)\) is completely determined by its entries in blocks \({\mathcal S}_{pq}(m)\) for \((p,q)\) restricted to \([1,m]\times[m+1,r]\cup[m+1,r]\times[1,m]\), called determining blocks. (Theorem 2.1). The authors also provide a method for constructing principal-ideal bands (Theorem 2.2). Matrix bands \({\mathcal S}^1\) and \({\mathcal S}^2\) in \(M_n(\mathbb{F})\) are called similar if there exists an invertible matrix \(X\) such that the map \(\phi_X\colon{\mathcal S}^1\to{\mathcal S}^2\) defined by \(\phi_X(S)=X^{-1}SX\) is a bijection from \({\mathcal S}^1\) to \({\mathcal S}^2\). The above decomposition (Theorem 2.2) is a canonical form with respect to similarity in the sense of Theorem 2.3.
Using the above Theorems, the authors give many properties about principal-ideal bands (Corollaries 3.1 to 3.5) and some related results, for example, Theorem 5.1: A finite operator band on a Banach space is triangularizable [the authors, Semigroup Forum 49, No. 2, 195-215 (1994; Zbl 0811.20061)].

MSC:

20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
47D03 Groups and semigroups of linear operators
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