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Inverse semirings and their lattice of congruences. (English) Zbl 0885.16027

The aim of this paper is to describe all inverse semirings \(\mathcal S\) having a modular (distributive, boolean) congruence lattice \(\text{Con}(\mathcal S)\). We shall fix the type \(\tau=(t,\text{ar})\) with \(t=(+,\cdot,-)\), \(\text{ar}(+)=\text{ar}(\cdot)=2\) and \(\text{ar}(-)=1\). An inverse semiring is a \(\tau\)-algebra \(\mathcal S=(S,\tau)\) satisfying the following axioms:
(1) \((S,+)\) is a commutative semigroup,
(2) \(x(y+z)=xy+xz\),
(3) \(x=(x-x)+x\), \(-x=-x+(x-x)\), where \(xy=x\cdot y\), \(xy+z=(xy)+z\) and \(x-y=x+(-y)\),
(4) \((x-x)+(y-y)=(x-x)(y-y)\).
By \(S(\mathcal S)\) we denote the set of all elements of an inverse semiring \(\mathcal S\). We put \(E(\mathcal S)=\{x\in S(\mathcal S):x=x+x\}\). The algebra \(\mathcal E(\mathcal S)=(E(\mathcal S),\tau)\) is an inverse subsemiring of \(\mathcal S\) which is a semilattice.
Theorem. Let \(\mathcal S\) be an inverse semiring. Then (i) \(\text{Con}(\mathcal S)\) is modular if and only if \(\mathcal E(\mathcal S)\) is a tree and \(\mathcal S\) has property (M); (ii) \(\text{Con}(\mathcal S)\) is distributive if and only if \(\mathcal E(\mathcal S)\) is a tree and \(\mathcal S\) has properties (M) and (D); (iii) \(\text{Con}(\mathcal S)\) is boolean if and only if \(\mathcal E(\mathcal S)\) is a locally finite tree and \(\mathcal S\) has property (B); (iv) \(\text{Con}(\mathcal S)\) is a chain if and only if \(\mathcal S\) is isomorphic to either \(\mathcal R\) or \({\mathcal R}^0\), where \(\mathcal R\) is a ring whose \(\text{Con}(\mathcal R)\) is a chain. Properties (M), (D) and (B): \(\mathcal S\) has property (M) if \(a+f=f\) for all \(a\), \(f\in S(\mathcal S)\), where \(f=0f<0a\); \(\mathcal S\) has property (D) if for each \(e\in E(\mathcal S)\) the lattice \(\text{Con}(\mathcal S_e)\) is distributive (\(\mathcal S_e\) is the inverse subsemiring of \(\mathcal S\) satisfying \(S(\mathcal S_e)=\{x\in S(\mathcal S):0x=e\}\)); \(\mathcal S\) has property (B) if \(\text{card }S(\mathcal S_e)>1\) implies that \(e\) is the zero of \(\mathcal E(\mathcal S)\) and \(\text{Con}(\mathcal S_e)\) is boolean.

MSC:

16Y60 Semirings
08A30 Subalgebras, congruence relations
06B15 Representation theory of lattices
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References:

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