Is the poset $(R, leq)$ in the semiring $R$ defined with respect to addition only?
A semiring $(R, +, cdot, leq)$ is said to be a partially ordered semiring if the relation $leq$ is a partial ordered relation on $R$ and $(R, +, cdot)$ satifies the following:
(i) additive monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $a+xleq a+y$ or $a+x=y$. Similarly, $x+aleq y+a$ or $x+a=y$ and
(ii) multiplicative monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $axleq ay$ and $xaleq ya$. I have a little confusion regarding the definition of the partial order relation $leq$ on $R$. I mean do we need to show that the relation is reflexive, anti-symmetric and transitive on $R$ with respect to addition only or, should it be a partial order relation with respect to multiplication also for $(R, leq)$ being a poset?
semiring
asked Nov 27 '18 at 18:47
gete
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A semiring $(R, +, cdot, leq)$ is said to be a partially ordered semiring if the relation $leq$ is a partial ordered relation on $R$ and $(R, +, cdot)$ satifies the following:
(i) additive monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $a+xleq a+y$ or $a+x=y$. Similarly, $x+aleq y+a$ or $x+a=y$ and
(ii) multiplicative monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $axleq ay$ and $xaleq ya$. I have a little confusion regarding the definition of the partial order relation $leq$ on $R$. I mean do we need to show that the relation is reflexive, anti-symmetric and transitive on $R$ with respect to addition only or, should it be a partial order relation with respect to multiplication also for $(R, leq)$ being a poset?
semiring
asked Nov 27 '18 at 18:47
gete
747
add a comment |
A semiring $(R, +, cdot, leq)$ is said to be a partially ordered semiring if the relation $leq$ is a partial ordered relation on $R$ and $(R, +, cdot)$ satifies the following:
(i) additive monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $a+xleq a+y$ or $a+x=y$. Similarly, $x+aleq y+a$ or $x+a=y$ and
(ii) multiplicative monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $axleq ay$ and $xaleq ya$. I have a little confusion regarding the definition of the partial order relation $leq$ on $R$. I mean do we need to show that the relation is reflexive, anti-symmetric and transitive on $R$ with respect to addition only or, should it be a partial order relation with respect to multiplication also for $(R, leq)$ being a poset?
semiring
asked Nov 27 '18 at 18:47
gete
747
A semiring $(R, +, cdot, leq)$ is said to be a partially ordered semiring if the relation $leq$ is a partial ordered relation on $R$ and $(R, +, cdot)$ satifies the following:
(i) additive monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $a+xleq a+y$ or $a+x=y$. Similarly, $x+aleq y+a$ or $x+a=y$ and
(ii) multiplicative monotonicity: $xleq y~forall ~x, yin R$ implies that there exists some $ain R$ such that $axleq ay$ and $xaleq ya$. I have a little confusion regarding the definition of the partial order relation $leq$ on $R$. I mean do we need to show that the relation is reflexive, anti-symmetric and transitive on $R$ with respect to addition only or, should it be a partial order relation with respect to multiplication also for $(R, leq)$ being a poset?
semiring
semiring
asked Nov 27 '18 at 18:47
gete
747
asked Nov 27 '18 at 18:47
gete
747
asked Nov 27 '18 at 18:47
gete
747
asked Nov 27 '18 at 18:47
gete
747
asked Nov 27 '18 at 18:47
gete
747
747
add a comment |
add a comment |
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