## 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

### References:

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