2010 Zoltan Esik, Andreas Maletti, "Simulation vs. Equivalence" arXiv
The semiring S is Noetherian if every subsemimodule of a finitely generated S-semimodule is finitely generated.
2011 February 10, Mikhail Aleksandrovich Shubin, Maxim Braverman, Spectral Theory and Geometric Analysis: International Conference in Honor of Mikhail Shubin's 65th Birthday, Spectral Theory and Geometric Analysis, July 29 - August 2, 2009, Northeastern University, Boston, MA, American Mathematical Soc., →ISBN, page 172:
... subsemimodule if it is closed under addition and multiplication by scalar coefficients. A subsemimodule V of a b-complete semimodule W is b-closed if V is closed under sums of any subsets of V that are bounded in W. A subsemimodule of a[…]
2013 April 18, Jonathan S. Golan, Semirings and their Applications, Springer Science & Business Media, →ISBN, page 155:
... subsemimodule of N and N is a subtractive subsemimodule of M then one sees immediately that N ' is a subtractive subsemimodule of M. If […] is a family of subtractive [resp. strong] subsemimodules of a left R - semimodule[…]
2012 March 14, Javed Ahsan, John N. Mordeson, Muhammad Shabir, Fuzzy Semirings with Applications to Automata Theory, Springer Science & Business Media, →ISBN, page 32:
... subsemimodule N of a right R-semimodule M is a subsemigroup ofM such that na G N for all n G N and a G R. By a right (left) ideal of R, we shall mean a subsemimodule of RR (RR). The word 'ideal' will always mean a two-sided ideal of R[…]
2013 May 24, Yves Achdou, Guy Barles, Hitoshi Ishii, Grigory L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications: Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna Tchou, Springer, →ISBN, page 268:
... subsemimodule is a b-complete semimodule. Note that quasisubsemimodules and ^-subsemimodules may fail to be ... subsemimodule of K.X/ consisting of all bounded mappings. If K is a b-complete semiring, then K.X/ and Kb .X/ are b-complete[…]
2013 March 14, Jonathan S. Golan, Semirings and Affine Equations over Them: Theory and Applications, Springer Science & Business Media, →ISBN, page 143:
... subsemimodule M of Rn is rational if and only if it is generated by a rational subset of the monoid ( R ” , · ) . It is immediately obvious that if M and M ' are rational subsemimodules of Rn then so is M + M ' . The following result[…]
2005, Grigoriĭ Lazarevich Litvinov, Viktor Pavlovich Maslov, Idempotent Mathematics and Mathematical Physics: International Workshop, February 3-10, 2003, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria, American Mathematical Soc., →ISBN, page 348:
... subsemimodule of M, • MB the largest Boolean subsemimodule of Ms B, • M1 the largest free subsemimodule of MB. [] REMARK 4.10. There is a direct sum analogue of Theorem 4. Its statement and proof are straightforward. 5. Examples In[…]
2003, K. P. Shum, Advances in Algebra: Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific, →ISBN, page 452:
... subsemimodule R of the direct product Ax A is called symmetric if (a, b) e R implies (b, a) e R c). If R is a symmetrie subsemimodule of Ax A, then there is a congruence relation p(R) on A as follows: x p(R) y if and only if there[…]
2012 December 6, Arto Salomaa, Matti Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer Science & Business Media, →ISBN, page 25:
... subsemimodule of AX* generated by the columns of H (r) is contained in a finitely generated stable subsemimodule of A**; (iii) r belongs to a finitely generated stable subsemimodule of A ** . PROOF . Suppose that ( i ) holds . By[…]
2009, Grigoriĭ Lazarevich Litvinov, S. N. Sergeev, Tropical and Idempotent Mathematics: International Workshop Tropical-07, Tropical and Idempotent Mathematics, August 25-30, 2007, Independent University of Moscow and Laboratory J.-V. Poncelet, American Mathematical Soc., →ISBN, page 7:
... subsemimodule of Rnmax. Every weakly independent generating family of M is obtained by picking exactly one non-zero element in each extremal ray. Proof. By [BSS07, Theorem 8], for a given subsemimodule M of Rnmax, any subset P of scaled[…]
1989, Société scientifique de Bruxelles, Annales: Sciences mathématiques, astronomiques et physiques. Series 1:
... subsemimodule of S, 0. Since S is 0-simple, this implies T = S, which is then a contradiction since TR = 0 and SR 0. Therefore 8R0. Since 8R is a R-subsemimodule of S, […]
1992, V. P. Maslov, Idempotent Analysis, American Mathematical Soc., →ISBN, page 71:
... subsemimodule generated by elements of the set A, where AC R3 is a two - dimensional disk of radius 1 which is orthogonal to the vector (1, 1, 1) […]. This subsemimodule on […]