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Minimal prime ideals in autometrized algebras. (English) Zbl 0814.06011
An autometrized \(\ell\)-algebra is a lattice-ordered commutative monoid \(A\) with an additional binary operation on \(A\) satisfying the properties of a metric. The author shows some connections between prime ideals of \(A\) and prime filters on the positive cone \(A^ +\), and between prime ideals and polars of \(A\), where \(A\) belongs to a special class of autometrized \(\ell\)-algebras (including Brouwerian algebras and commutative \(\ell\)-groups). Many of the properties of minimal prime ideals and polars are strengthened for dually residuated \(\ell\)- semigroups, which include Boolean algebras and commutative \(\ell\)-groups.

06F05 Ordered semigroups and monoids
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[1] Hansen, M. E.: Filets in autometrized algebras; in preparation.
[2] Rachůnek, J.: Prime ideals in autometrized algebras. Czechoslovak Math. J. 37 (112) (1987), 65-69. · Zbl 0692.06007
[3] Rachůnek, J.: Polars in autometrized algebras. Czechoslovak Math. J. 39 (114) (1989), 681-685. · Zbl 0705.06010
[4] Rachůnek, J.: Regular ideals in autometrized algebra. Math. Slovaca 40 (1990), 117-122. · Zbl 0738.06014
[5] Swamy, K. L. N.: A general theory of autometrized algebras. Math. Ann. 157 (1964), 65-74. · Zbl 0135.02602
[6] Swamy, K. L. N.: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105-114. · Zbl 0138.02104
[7] Swamy, K. L. N. and Rao, N. P.: Ideals in autometrized algebras. J. Austral. Math. Soc. (Ser. A) 24 (1977), 362-374. · Zbl 0427.06006
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