##
**On the structure of semifields and lattice-ordered groups.**
*(English)*
Zbl 0896.12001

The authors continue their study of the radical theory of semifields in their previous article [Commun. Algebra 20, 2419-2458 (1992; Zbl 0771.16016)]. They prove several interesting structure theorems for semisimple classes and radical classes of semifields that they apply to lattice ordered groups.

We quote only a very small number of these results. A semifield is a universal algebra \(\langle A;+, \cdot\rangle\) where \(\langle A;\cdot\rangle\) is a group with multiplicative identity \(e\), \(\langle A;+\rangle\) is a semigroup and \(\cdot\) left and right distributes over \(+\). The kernels of homomorphisms are determined by their \(e\)-classes which are normal subgroups \(K\) of \(\langle A;\cdot\rangle\) for which for any \(s,t \in A\), \(h\in K\), \(s+t\in K\) implies \(s+th\in K\). Addition in \(A/K\) is idempotent iff \(K\) is a subsemifield of \(\langle A;+,\cdot\rangle\). Let \({\mathfrak S}^*\) be the class of all semifields and let \(\mathfrak G\) be the class of all groups. The authors’ Kurosh-Amitsur radical theory for semifields has been developed within the category \({\mathfrak S}^* \cup \mathfrak G\) where the morphisms \(\phi :A \rightarrow B\) satisfy \(\phi \) is a semifield (group) homomorphism if \(A,B \in {\mathfrak S}^*\) (\(A,B \in \mathfrak G\)) and \(\phi \) preserves multiplication if \(A\in \mathfrak G , B\in \mathfrak S^*\). Radical classes and the corresponding semisimple classes are defined and the theory contains the Kurosh-Amitsur theory for groups. It is shown that under some natural conditions on a class \(M\), the class of all subdirect products of members of \(M\) is semisimple. The authors also study simple and subdirectly irreducible semifields.

Theorem 1: Let \(A\) be a semifield and let \(K\) be a kernel of \(A\) which is also a semifield. Set \(B=A/K\). Then \(\langle A;\cdot\rangle\) is isomorphic to a Schreier extension \(B\circ K\) with the property that the automorphism systems \({\alpha }^b\) are actually semifield automorphisms of \(K\).

The authors give a construction of semifields similar to the theorem which leads in the case when \(K\) is simple to a subdirectly irreducible semifield \(A\) whose heart is \(K\).

A lattice ordered group can be viewed as a semifield whose additive structure is a semilattice. The kernels of such semifields are semifields and correspond to the convex normal subgroups. Theorem 2: If \(R\) is a kernel in a commutative lattice ordered group \(A\), then \(R\) is closed in \(A\). If \(A\) is a complete lattice ordered group, then every kernel of \(A\) is a direct factor.

We quote only a very small number of these results. A semifield is a universal algebra \(\langle A;+, \cdot\rangle\) where \(\langle A;\cdot\rangle\) is a group with multiplicative identity \(e\), \(\langle A;+\rangle\) is a semigroup and \(\cdot\) left and right distributes over \(+\). The kernels of homomorphisms are determined by their \(e\)-classes which are normal subgroups \(K\) of \(\langle A;\cdot\rangle\) for which for any \(s,t \in A\), \(h\in K\), \(s+t\in K\) implies \(s+th\in K\). Addition in \(A/K\) is idempotent iff \(K\) is a subsemifield of \(\langle A;+,\cdot\rangle\). Let \({\mathfrak S}^*\) be the class of all semifields and let \(\mathfrak G\) be the class of all groups. The authors’ Kurosh-Amitsur radical theory for semifields has been developed within the category \({\mathfrak S}^* \cup \mathfrak G\) where the morphisms \(\phi :A \rightarrow B\) satisfy \(\phi \) is a semifield (group) homomorphism if \(A,B \in {\mathfrak S}^*\) (\(A,B \in \mathfrak G\)) and \(\phi \) preserves multiplication if \(A\in \mathfrak G , B\in \mathfrak S^*\). Radical classes and the corresponding semisimple classes are defined and the theory contains the Kurosh-Amitsur theory for groups. It is shown that under some natural conditions on a class \(M\), the class of all subdirect products of members of \(M\) is semisimple. The authors also study simple and subdirectly irreducible semifields.

Theorem 1: Let \(A\) be a semifield and let \(K\) be a kernel of \(A\) which is also a semifield. Set \(B=A/K\). Then \(\langle A;\cdot\rangle\) is isomorphic to a Schreier extension \(B\circ K\) with the property that the automorphism systems \({\alpha }^b\) are actually semifield automorphisms of \(K\).

The authors give a construction of semifields similar to the theorem which leads in the case when \(K\) is simple to a subdirectly irreducible semifield \(A\) whose heart is \(K\).

A lattice ordered group can be viewed as a semifield whose additive structure is a semilattice. The kernels of such semifields are semifields and correspond to the convex normal subgroups. Theorem 2: If \(R\) is a kernel in a commutative lattice ordered group \(A\), then \(R\) is closed in \(A\). If \(A\) is a complete lattice ordered group, then every kernel of \(A\) is a direct factor.

Reviewer: A.A.Iskander (Lafayette)

### MSC:

12K10 | Semifields |

06F15 | Ordered groups |

20F60 | Ordered groups (group-theoretic aspects) |

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

### Citations:

Zbl 0771.16016
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\textit{H. J. Weinert} and \textit{R. Wiegandt}, Period. Math. Hung. 32, No. 1--2, 129--147 (1996; Zbl 0896.12001)

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### References:

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[4] | L.Rédei,Algebra, Geest & Portig K.-G., Leipzig, 1959. |

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[8] | H.J.Weinert, R.Wiegandt, A Kurosh-Amitsur radical theory for proper semifields,Comm. in Algebra 20 (1992), 2419–2458. · Zbl 0771.16016 |

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