##
**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.

(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.

Reviewer: František Šik (Brno)

### MSC:

16Y60 | Semirings |

08A30 | Subalgebras, congruence relations |

06B15 | Representation theory of lattices |

### Keywords:

inverse semirings; modular congruence lattices; distributive congruence lattices; Boolean congruence lattices
PDF
BibTeX
XML
Cite

\textit{B. Pondělíček}, Czech. Math. J. 46, No. 3, 513--522 (1996; Zbl 0885.16027)

### References:

[1] | H. Mitsch: Semigroups and their lattice of congruences. Semigroup Forum 26 (1983), 1-63. · Zbl 0513.20047 |

[2] | V. N. Salij: To the theory of inverse semirings (Russian). Izv. vuzov. Matem. (1969), 52-60. |

[3] | B. B. Kovalenko: On the theory of generalized modules (Russian). Izdat. Saratov. Univ. Saratov, Studies in algebra (1977), no. No. 5, 30-43. |

[4] | G. Grätzer: Lattice theory, first concepts and distributive lattices. vol. , San Francisco, 1971. · Zbl 0232.06001 |

[5] | T. Tamura: Commutative semigroups whose lattice of congruences is a chain. Bull. Soc. Math. France 97 (1969), 369-380. · Zbl 0191.01705 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.